RedcrabX – Geometry Functions

Reference for the Geometry panel | Lines and Curves & Angle groups

Related Physics helper: centforce(...=..., ...=..., ...=...) in the Physics panel computes centrifugal-force quantities via F = m·v²/r (solve any missing one of F, m, v, r).

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Lines and Curves
Triangle
Angular
Circular
Spherical
Cylinder
Pyramid
Cone
Angle Group
Quick Reference

Overview

The Geometry panel provides analytic-geometry helpers for 2D coordinate calculations. All functions return a named result list when used with the = display operator (e.g. a := p2dist(0,0,3,4)=), so each computed quantity appears on its own labelled row with its unit. Individual results can be accessed via dot notation: a.c, a.b, a.alpha, etc.

Functions are located in the Geometry panel under the Lines and Curves group (coordinate & curve helpers) and the Angle group (conversion, normalization, and angle relations).

Point-to-Point Distance – `p2dist`

Syntax

p2dist(Ax, Ay, Bx, By)

Description

Computes the geometric distance between two points A(Ax, Ay) and B(Bx, By) in the 2D coordinate plane using the Pythagorean theorem. The function also returns the horizontal and vertical components and the angle of the line relative to the x-axis.

Y | B(Bx,By) | /| | c / | a = |By-Ay| | / | | / α | | A(Ax,Ay)----- | b = |Bx-Ax| +--------------------→ X

Formulas

Quantity	Formula	Unit
`b` – horizontal distance	\|Bx − Ax\|	–
`a` – vertical distance	\|By − Ay\|	–
`c` – total distance A↔B	√(a² + b²)	–
`alpha` – angle α	atan(a / b) × 180/π	°

Result List

Label	Key	Description
Distance A↔B (c)	`.c`	Direct distance between A and B
Distance X (b)	`.b`	Horizontal component \|Bx − Ax\|
Distance Y (a)	`.a`	Vertical component \|By − Ay\|
Angle α	`.alpha`	Angle of AB relative to the x-axis in degrees

Examples

r := p2dist(0, 0, 3, 4)=

Distance A↔B (c)    5
Distance X (b)      3
Distance Y (a)      4
Angle α           53.13 °

Access individual values:

r.c      → 5        (total distance)
r.b      → 3        (horizontal)
r.a      → 4        (vertical)
r.alpha  → 53.13°   (angle)

p2dist(1, 2, 4, 6)=

Distance A↔B (c)    5
Distance X (b)      3
Distance Y (a)      4
Angle α           53.13 °

Note: The unit of c, b, and a is the same as the unit of the coordinates you supply (pixels, metres, mm, etc.). RedcrabX treats the numbers as dimensionless; apply your own scale factor if needed.

Point-to-Line Distance – `p2linedist`

Syntax

p2linedist(Ax, Ay, Bx, By, Px, Py)

Description

Computes the shortest (perpendicular) Euclidean distance from point P(Px, Py) to the infinite line passing through A(Ax, Ay) and B(Bx, By). The function also returns the foot point F (the closest point on the line to P) and the length of segment AB.

Formulas

Let AB be the direction vector of the line: dx = Bx−Ax, dy = By−Ay.

Quantity	Formula	Unit
`d` – shortest distance	\|(dx·(Py−Ay) − dy·(Px−Ax))\| / \|AB\|	–
`t` – projection parameter	((Px−Ax)·dx + (Py−Ay)·dy) / (dx²+dy²)	–
`Fx` – foot point x	Ax + t·dx	–
`Fy` – foot point y	Ay + t·dy	–
`\|AB\|` – line length	√(dx² + dy²)	–

Note: The line is treated as infinite. If t < 0 the foot point lies before A; if t > 1 it lies beyond B. To get the distance to the segment AB instead, clamp t to [0, 1] and recompute.

Result List

Label	Key	Description
Shortest distance (d)	`.d`	Perpendicular distance from P to the line
Foot point X (Fx)	`.fx`	x-coordinate of the closest point on the line
Foot point Y (Fy)	`.fy`	y-coordinate of the closest point on the line
Line length \|AB\|	`.ab`	Length of the reference segment A→B

Examples

r := p2linedist(0, 0, 10, 0, 3, 4)=

Line along x-axis from (0,0) to (10,0); point P at (3,4). The perpendicular foot is directly below P on the x-axis.

Shortest distance (d) 4
Foot point X (Fx)      3
Foot point Y (Fy)      0
Line length |AB|       10

r := p2linedist(0, 0, 3, 4, 1, 3)=

Shortest distance (d) 1.4
Foot point X (Fx)      1.68
Foot point Y (Fy)      2.24
Line length |AB|       5

Access individual values:

r.d   → shortest distance
r.fx  → foot point x
r.fy  → foot point y
r.ab  → segment length |AB|

Line-to-Line Angle – `l2angle`

Syntax

l2angle(Ax, Ay, Bx, By, Cx, Cy, Dx, Dy)

Description

Computes the acute intersection angle α between two infinite lines in the 2D plane. Line 1 is defined by points A(Ax, Ay) and B(Bx, By); Line 2 is defined by points C(Cx, Cy) and D(Dx, Dy). The result is always in the range 0° … 90°.

Y | B | / | / Line 1 | / | A | D | / | / Line 2 α = acute angle between lines | / | C +-----------------------------→ X

Formula

Let d₁ = (Bx−Ax, By−Ay) and d₂ = (Dx−Cx, Dy−Cy) be the direction vectors.

Quantity	Formula	Unit
`alpha` – intersection angle α	acos(\|d₁·d₂\| / (\|d₁\|·\|d₂\|)) × 180/π	°

Note: The absolute value of the dot product gives the acute angle. Parallel lines return 0°; perpendicular lines return 90°. The position of the lines in the plane does not matter — only their direction vectors.

Result List

Label	Key	Description
Intersection angle α	`.alpha`	Acute angle between the two lines in degrees

Examples

l2angle(0,0,1,0, 0,0,1,1)=

Intersection angle α 45 °

r := l2angle(0,0,3,0, 0,0,0,5)=

Horizontal line vs. vertical line → 90°.

Intersection angle α 90 °

l2angle(0,0,1,1, 0,0,-1,1)=

Two lines at 45° to the x-axis, symmetric about y-axis → 90°.

Intersection angle α 90 °

Access the value via dot notation:

r := l2angle(0,0,1,0, 0,0,1,1)
r.alpha  → 45°

2D Vector Angle – `v2angle`

Syntax

v2angle(Ax, Ay, Bx, By)

Description

Computes the angle between two 2D vectors A(Ax, Ay) and B(Bx, By) using the dot-product formula. The result is in the range 0° … 180°. Unlike l2angle, this function considers direction — antiparallel vectors give 180°.

Formula

Quantity	Formula	Unit
`alpha` – angle α	acos((A·B) / (\|A\|·\|B\|)) × 180/π	°
`dot` – dot product	Ax·Bx + Ay·By	–
`lenA` – \|A\|	√(Ax² + Ay²)	–
`lenB` – \|B\|	√(Bx² + By²)	–

Result List

Label	Key	Description
Angle α	`.alpha`	Angle between the two vectors in degrees
Dot product	`.dot`	Scalar dot product A·B
\|A\|	`.lenA`	Magnitude of vector A
\|B\|	`.lenB`	Magnitude of vector B

Examples

v2angle(1, 0, 0, 1)=

Angle α       90 °
Dot product 0
|A|           1
|B|           1

v2angle(1, 0, -1, 0)=

Angle α       180 °
Dot product −1
|A|           1
|B|           1

r := v2angle(3, 0, 3, 3)
r.alpha   → 45°
r.dot     → 9

3D Vector Angle – `v3angle`

Syntax

v3angle(Ax, Ay, Az, Bx, By, Bz)

Description

Computes the angle between two 3D vectors A(Ax, Ay, Az) and B(Bx, By, Bz) using the dot-product formula. The result is in the range 0° … 180°.

Formula

Quantity	Formula	Unit
`alpha` – angle α	acos((A·B) / (\|A\|·\|B\|)) × 180/π	°
`dot` – dot product	Ax·Bx + Ay·By + Az·Bz	–
`lenA` – \|A\|	√(Ax² + Ay² + Az²)	–
`lenB` – \|B\|	√(Bx² + By² + Bz²)	–

Result List

Label	Key	Description
Angle α	`.alpha`	Angle between the two vectors in degrees
Dot product	`.dot`	Scalar dot product A·B
\|A\|	`.lenA`	Magnitude of vector A
\|B\|	`.lenB`	Magnitude of vector B

Examples

v3angle(1, 0, 0, 0, 1, 0)=

Angle α       90 °
Dot product 0
|A|           1
|B|           1

v3angle(1, 1, 0, 1, 0, 0)=

Angle α       45 °
Dot product 1
|A|           1.414…
|B|           1

r := v3angle(0, 0, 1, 1, 0, 0)
r.alpha   → 90°
r.dot     → 0

Triangle Angles from Sides – `abcangles`

Syntax

abcangles(a, b, c)

Description

Computes triangle interior angles from side lengths a, b, and c using the law of cosines. Angles are returned in degrees: α opposite side a, β opposite side b, and γ opposite side c.

Inputs must satisfy a > 0, b > 0, c > 0 and the triangle inequality.

Formulas

Quantity	Formula	Unit
`alpha`	acos((b² + c² − a²) / (2bc)) × 180/π	°
`beta`	acos((a² + c² − b²) / (2ac)) × 180/π	°
`gamma`	180° − α − β	°

Result List

Label	Key	Description
Angle α	`.alpha`	Angle opposite side a
Angle β	`.beta`	Angle opposite side b
Angle γ	`.gamma`	Angle opposite side c

Examples

abcangles(3, 4, 5)=

Angle α       36.8699 °
Angle β        53.1301 °
Angle γ     90 °

r := abcangles(7, 8, 9)
r.alpha   → 48.19°
r.beta    → 58.41°
r.gamma   → 73.40°

Polygon Angle Sum and Single Angles – `polyangles`

Syntax

polyangles(n)

Description

Computes angle values for a regular polygon with n sides: the sum of interior angles, one interior angle, and one exterior angle.

Input requirement: n must be an integer with n ≥ 3.

Formulas

Quantity	Formula	Unit
`sum` – interior angle sum	(n − 2) × 180	°
`interior` – one interior angle	sum / n	°
`exterior` – one exterior angle	360 / n	°

Result List

Label	Key	Description
Number of sides	`.n`	Input side count
Interior angle sum	`.sum`	Total of all interior angles
Single interior angle	`.interior`	Interior angle of each corner (regular polygon)
Single exterior angle	`.exterior`	Exterior turn angle per corner

Examples

polyangles(5)=

Number of sides        5
Interior angle sum 540 °
Single interior      108 °
Single exterior      72 °

p := polyangles(8)
p.sum       → 1080°
p.interior  → 135°
p.exterior  → 45°

Equilateral Triangle Solver – `eqtri`

Syntax

eqtri(a=...)
eqtri(h=...)
eqtri(u=...)
eqtri(A=...)

Description

Solves a perfectly symmetric triangle (equilateral triangle) from one known quantity. You can provide side length a, height h, perimeter u, or area A. The function returns all four values.

Formulas

Quantity	Formula	Unit
`h`	a × √3 / 2	–
`u`	3a	–
`A`	a² × √3 / 4	–
`a` from `h`	2h / √3	–
`a` from `u`	u / 3	–
`a` from `A`	√(4A / √3)	–

Result List

Label	Key	Description
Side length	`.a`	Equal side length
Height	`.h`	Triangle altitude
Perimeter	`.u`	Total boundary length
Area	`.A`	Triangle area

Examples

eqtri(a=6)=

Side length     6
Height         5.19615
Perimeter      18
Area           15.58846

t := eqtri(A=20)
t.a   → 6.798
t.h   → 5.887
t.u   → 20.394
t.A   → 20

Isosceles Triangle Solver – `isotri`

Syntax

isotri(...=..., ...=...)

Description

Solves an isosceles triangle from two named inputs with this pair rule: one input from (a | h) and one input from (b | h | A | alpha). The function returns the full set: base a, side b, height h, perimeter P, area A, base angle alpha, and apex angle gamma.

Formulas

Quantity	Formula	Unit
`P` – perimeter	a + 2b	–
`A` – area	a × h / 2	–
`alpha` – base angle	atan(h / (a/2)) × 180/π	°
`gamma` – apex angle	180° − 2 × alpha	°

Result List

Label	Key	Description
Base a	`.a`	Base length
Side b	`.b`	Equal side length
Height h	`.h`	Height to base a
Perimeter P	`.P`	Total boundary length
Area A	`.A`	Triangle area
Angle alpha	`.alpha`	Base angle in degrees
Angle gamma	`.gamma`	Apex angle in degrees

Examples

isotri(a=10, h=6)=

Base a           10
Side b           7.81025
Height h        6
Perimeter P     25.6205
Area A           30
Angle alpha    50.1944 °
Angle gamma    79.6111 °

t := isotri(h=6, alpha=40)
t.a      → 14.301
t.b      → 9.334
t.P      → 32.969
t.A      → 42.902
t.gamma  → 100

Right Triangle Solver – `rtri`

Syntax

rtri(...=..., ...=...)

Description

Solves a right triangle from one side (a, b, or c) and one acute angle alpha. The function returns: catheti a, b, hypotenuse c, altitude to hypotenuse h, area A, perimeter P, hypotenuse segments p, q, and angles alpha, beta.

Formulas

Quantity	Formula	Unit
`h`	a × b / c	–
`A`	a × b / 2	–
`P`	a + b + c	–
`p`	a² / c	–
`q`	b² / c	–
`beta`	90° − alpha	°

Result List

Label	Key	Description
Cathetus a	`.a`	Leg opposite angle alpha
Cathetus b	`.b`	Leg adjacent to angle alpha
Hypotenuse c	`.c`	Longest side
Height h	`.h`	Altitude to hypotenuse
Area A	`.A`	Triangle area
Perimeter P	`.P`	Total boundary length
Segment p	`.p`	Hypotenuse segment next to cathetus a
Segment q	`.q`	Hypotenuse segment next to cathetus b
Angle alpha	`.alpha`	Given acute angle in degrees
Angle beta	`.beta`	Other acute angle in degrees

Examples

rtri(a=5, alpha=30)=

Cathetus a      5
Cathetus b      8.66025
Hypotenuse c 10
Height h        4.33013
Area A          21.6506
Perimeter P    23.6603
Segment p       2.5
Segment q       7.5
Angle alpha 30 °
Angle beta    60 °

rt := rtri(c=12, alpha=40)
rt.a     → 7.713
rt.b     → 9.193
rt.h     → 5.907
rt.A     → 35.443
rt.P     → 28.906
rt.beta  → 50

Pythagoras Right-Triangle Solver – `pythag`

Syntax

pythag(...=..., ...=...)

Description

Fully solves a right triangle from two known sides using Pythagoras. Supported input pairs are (a,b), (a,c), or (b,c). The function returns: catheti a, b, hypotenuse c, height to hypotenuse h, area A, perimeter P, hypotenuse segments p, q, and angles alpha, beta.

Formulas

Quantity	Formula	Unit
`c` from `a,b`	√(a² + b²)	–
`b` from `a,c`	√(c² − a²)	–
`a` from `b,c`	√(c² − b²)	–
`h`	a × b / c	–
`A`	a × b / 2	–
`P`	a + b + c	–
`p`	a² / c	–
`q`	b² / c	–
`alpha`	asin(a / c) × 180/π	°
`beta`	90° − alpha	°

Result List

Label	Key	Description
Cathetus a	`.a`	Leg a
Cathetus b	`.b`	Leg b
Hypotenuse c	`.c`	Hypotenuse
Height h	`.h`	Altitude to hypotenuse
Area A	`.A`	Triangle area
Perimeter P	`.P`	Total boundary length
Segment p	`.p`	Hypotenuse segment near a
Segment q	`.q`	Hypotenuse segment near b
Angle alpha	`.alpha`	Acute angle opposite a
Angle beta	`.beta`	Other acute angle

Examples

pythag(a=3, b=4)=

Cathetus a      3
Cathetus b      4
Hypotenuse c 5
Height h        2.4
Area A          6
Perimeter P    12
Segment p       1.8
Segment q       3.2
Angle alpha 36.8699 °
Angle beta    53.1301 °

rt2 := pythag(a=5, c=13)
rt2.b     → 12
rt2.h     → 4.615
rt2.A     → 30
rt2.P     → 30
rt2.alpha → 22.62
rt2.beta  → 67.38

Isosceles Right-Triangle Solver – `isortri`

Syntax

isortri(a=...)
isortri(b=...)
isortri(h=...)
isortri(A=...)

Description

Solves an isosceles right triangle from one named input. Inputs: base a (hypotenuse), side b (cathetus), height h to base, or area A. Returned values: base a, side b, height h, perimeter P, and area A.

Formulas

Quantity	Formula	Unit
`b` from `a`	a / √2	–
`h` from `a`	a / 2	–
`A` from `a`	a² / 4	–
`a` from `b`	b × √2	–
`h` from `b`	b / √2	–
`a` from `h`	2h	–
`a` from `A`	2√A	–
`P`	a + 2b	–

Result List

Label	Key	Description
Base a	`.a`	Hypotenuse
Side b	`.b`	Equal cathetus length
Height h	`.h`	Altitude to base a
Perimeter P	`.P`	Total boundary length
Area A	`.A`	Triangle area

Examples

isortri(a=10)=

Base a           10
Side b           7.07107
Height h        5
Perimeter P     24.1421
Area A           25

s := isortri(A=36)
s.a  → 12
s.b  → 8.485
s.h  → 6
s.P  → 28.971
s.A  → 36

Triangle Area from Base and Height – `triarea`

Syntax

triarea(base, height)
triangarea(a, b, alpha)
triarea2ang(base, alpha, beta)
triheron(a, b, c)

Description

Computes the area of a triangle from base and height, from two sides with included angle alpha, from base with two base angles alpha and beta, or from all three sides using Heron's formula. For triangarea(a,b,alpha), alpha follows the toolbar angle mode (DEG/RAD). For triarea2ang(base,alpha,beta), both angles follow toolbar angle mode (DEG/RAD).

Formula

Quantity	Formula	Unit
`A` from base/height	base × height / 2	–
`A` from two sides and angle	1/2 × a × b × sin(alpha)	–
`A` from base and two angles	base² × sin(alpha) × sin(beta) / (2 × sin(alpha+beta))	–
`A` from three sides (Heron)	√(s(s-a)(s-b)(s-c)), s=(a+b+c)/2	–

Examples

triarea(10, 6)         → 30
triarea(4.2, 3.5)      → 7.35
DEG: triangarea(7, 9, 30)   → 15.75
RAD: triangarea(7, 9, π/6)  → 15.75
DEG: triarea2ang(10, 40, 60)  → 30.687
triheron(3, 4, 5)  → 6

Triangle Area from Three Sides (Heron) – `triheron`

Syntax

triheron(a, b, c)

Description

Computes triangle area from three side lengths using Heron's formula. Requires a,b,c > 0 and the triangle inequality.

Formula

Quantity	Formula	Unit
`s` – semiperimeter	(a+b+c)/2	–
`A` – area	√(s(s-a)(s-b)(s-c))	–

Examples

triheron(3, 4, 5)      → 6
triheron(5, 5, 6)      → 12

Triangle Area from Base and Two Angles – `triarea2ang`

Syntax

triarea2ang(base, alpha, beta)
triangarea2ang(base, alpha, beta)   // alias

Description

Computes the triangle area from base length and two base angles. Angles alpha and beta follow the toolbar angle mode (DEG/RAD).

Formula

Quantity	Formula	Unit
`A`	base² × sin(alpha) × sin(beta) / (2 × sin(alpha+beta))	–

Examples

triarea2ang(10, 40, 60)      → 30.687
triangarea2ang(10, 40, 60)   → 30.687

Square Solver – `square`

Syntax

square(name=value)
quadrat(name=value)   // alias

Description

Solves a square from one known value. Supported input names: seite, diagonal, umfang, flaeche (or short forms a, d, U, A).

Result List

Label	Key	Description
Seitenlänge a	`.a`	Side length
Diagonale d	`.d`	Diagonal
Umfang U	`.U`	Perimeter
Fläche A	`.A`	Area
Innenkreisradius ri	`.ri`	Inradius
Innenkreisfläche Ai	`.Ai`	Incircle area
Umkreisradius ru	`.ru`	Circumradius
Umkreisfläche Au	`.Au`	Circumcircle area

Examples

sq := square(seite=5)=
sq.a  → 5
sq.d  → 7.071
sq.U  → 20
sq.A  → 25

square(flaeche=25)=
quadrat(diagonal=10)=

Golden Rectangle Solver – `goldrect`

Syntax

goldrect(name=value)
goldenrect(name=value)      // alias
goldenrectangle(name=value) // alias

Description

Solves a golden rectangle from one known value. Supported input names: a, b, d, U, A and aliases laenge, breite, diagonal, umfang, flaeche. The side ratio is fixed to φ = (1+√5)/2 ≈ 1.618.

Result List

Label	Key	Description
Seitenlänge a	`.a`	Long side
Seitenlänge b	`.b`	Short side
Diagonale d	`.d`	Diagonal
Umfang U	`.U`	Perimeter
Fläche A	`.A`	Area
Verhältnis φ	`.phi`	Golden ratio (~1.618)

Examples

gr := goldrect(a=10)=
gr.a   → 10
gr.b   → 6.18034
gr.d   → 11.7557
gr.U   → 32.3607
gr.A   → 61.8034
gr.phi → 1.61803

goldrect(A=40)=
goldenrectangle(d=12)=

Parallelogram Solver – `para`

Syntax

para(a, b, alpha)
parallelogram(a, b, alpha)   // alias
parallelogramm(a, b, alpha)  // alias

Description

Computes a classical parallelogram from side lengths a, b and included angle alpha in degrees.

Result List

Label	Key	Description
Fläche A	`.A`	Area
Umfang P	`.P`	Perimeter
Höhe ha	`.ha`	Height to side a
Höhe hb	`.hb`	Height to side b
Diagonale e	`.e`	Long diagonal
Diagonale f	`.f`	Short diagonal
Winkel β	`.β`	Second interior angle (β = 180° - α)

Examples

p := para(8, 5, 60)=
p.A  → 34.641
p.P  → 26
p.ha → 4.330
p.hb → 6.928
p.e  → 11.357
p.f  → 7
p.β  → 120°

Parallelogram Area from Base and Height – `paraaera`

Syntax

paraaera(base, height)
parallelarea(base, height)   // alias

Description

Computes the area of a parallelogram directly from base length and corresponding height.

Formula

Quantity	Formula	Unit
`A` – area	base × height	–

Examples

paraaera(8, 5)   → 40
parallelarea(12.5, 3.2)   → 40

Rectangle Solver – `rect`

Syntax

rect(name=value, name=value)
rectangle(name=value, name=value)   // alias
rechteck(name=value, name=value)    // alias

Description

Solves a classic rectangle from two named inputs. Allowed pair rule: one of (l|b) and one of (b|d|A|U). Supports aliases like length/laenge, breite, diag/diagonal, flaeche/area, umfang/perimeter.

Result List

Label	Key	Description
Länge l	`.l`	Rectangle length
Breite b	`.b`	Rectangle width
Diagonale d	`.d`	Diagonal
Umfang U	`.U`	Perimeter
Fläche A	`.A`	Area

Examples

r := rect(l=8, b=5)=
r.l  → 8
r.b  → 5
r.d  → 9.434
r.U  → 26
r.A  → 40

rect(l=8, d=10)=
rechteck(b=6, U=20)=

Slope / Gradient – `slope`

Syntax

slope(rise, run)

Description

Computes the slope (Steigung) of an incline from the vertical rise (Anstieg) and the horizontal run (horizontale Strecke). All three common representations are returned: the dimensionless slope m, the angle α in degrees, and the road-engineering grade in percent.

Formulas

Quantity	Formula	Unit
`m` – slope	rise / run	–
`alpha` – angle α	atan(rise / run) × 180/π	°
`grade` – Gefälle / Steigung	m × 100	%
`rise`	vertical height (input)	–
`run`	horizontal distance (input)	–

Result List

Label	Key	Description
Slope m	`.m`	Dimensionless slope (rise/run)
Angle α	`.alpha`	Inclination angle in degrees
Grade	`.grade`	Slope in percent (road / roof convention)
Rise	`.rise`	Vertical component (input)
Run	`.run`	Horizontal component (input)

Examples

slope(1, 10)=

Slope m       0.1
Angle α       5.711 °
Grade         10 %
Rise          1
Run           10

slope(3, 4)=

Slope m       0.75
Angle α       36.87 °
Grade         75 %
Rise          3
Run           4

r := slope(1, 10)
r.m       → 0.1
r.alpha   → 5.711
r.grade   → 10

Point-to-Point Slope – `p2slope`

Syntax

p2slope(Ax, Ay, Bx, By)

Description

Computes the slope (Steigung) of the line through two coordinate points A(Ax, Ay) and B(Bx, By). All three common representations are returned: the dimensionless slope m, the inclination angle α in degrees, and the grade in percent. Δy and Δx are also returned as rise and run. The function throws an error when A and B share the same x-coordinate (vertical line).

Formulas

Quantity	Formula	Unit
`m` – slope	(By − Ay) / (Bx − Ax)	–
`alpha` – angle α	atan(m) × 180/π	°
`grade`	m × 100	%
`rise` – Δy	By − Ay	–
`run` – Δx	Bx − Ax	–

Result List

Label	Key	Description
Slope m	`.m`	Dimensionless slope
Angle α	`.alpha`	Inclination angle in degrees
Grade	`.grade`	Slope in percent
Rise Δy	`.rise`	Vertical distance By−Ay
Run Δx	`.run`	Horizontal distance Bx−Ax

Examples

p2slope(0, 0, 10, 1)=

Slope m       0.1
Angle α       5.711 °
Grade         10 %
Rise Δy       1
Run Δx        10

p2slope(2, 3, 6, 7)=

Slope m       1
Angle α       45 °
Grade         100 %
Rise Δy       4
Run Δx        4

r := p2slope(0, 0, 10, 1)
r.m       → 0.1
r.alpha   → 5.711
r.grade   → 10

Circle Solver – `circle`

Syntax

circle(...=...)
circ(...=...)     // alias
kreis(...=...)    // alias

Description

Computes all main circle dimensions from exactly one named input. Supported input names are r (radius), d (diameter), U (perimeter/circumference), and A (area).

Formulas

Quantity	Symbol	Formula	Unit
Radius	`r`	d/2, U/(2π), or √(A/π)	–
Diameter	`d`	2r	–
Perimeter	`U`	2πr	–
Area	`A`	πr²	–

Result List

Label	Key	Description
Radius r	`.r`	Circle radius
Diameter d	`.d`	Circle diameter
Perimeter U	`.U`	Circumference
Area A	`.A`	Disk area

Examples

circle(r=5)=

Radius r        5
Diameter d     10
Perimeter U 31.416
Area A          78.54

circle(d=10)=

circle(U=31.416)=

circle(A=78.54)=

Circular Sector Solver – `sect`

Syntax

sect(r, ...=...)
sector(r, ...=...)   // alias

Description

Computes circular-sector dimensions from radius r and one named parameter. Supported named inputs are: alpha (angle in degrees), A (area), P (perimeter), b (arc length), or s (chord).

Result List

Label	Key	Description
Angle α	`.α`	Central angle in degrees
Area A	`.A`	Sector area
Perimeter P	`.P`	Boundary length (2r + b)
Arc length b	`.b`	Arc of the sector
Chord s	`.s`	Straight edge between arc endpoints
Centroid S	`.S`	Distance of centroid from circle center on angle bisector
Height h	`.h`	Sagitta of the sector arc

Examples

sect(5, alpha=90)=

Angle α       90 °
Area A         19.635
Perimeter P 17.854
Arc length b 7.854
Chord s        7.071
Centroid S     3.001
Height h       1.464

sect(5, A=19.635)=

sect(5, P=17.854)=

sect(5, b=7.854)=

sect(5, s=7.071)=

Circular Segment Solver – `segm`

Syntax

segm(...=..., ...=...)
segment(...=..., ...=...)   // alias

Description

Computes circular-segment (circular-cap) dimensions from two named inputs. Allowed input pairs are: (r,alpha), (r,s), (h,alpha), or (h,s). Here r is radius, h is cap height, and s is chord length. The angle alpha is in degrees.

Result List

Label	Key	Description
Angle α	`.α`	Central angle in degrees
Height h	`.h`	Segment height (sagitta)
Area A	`.A`	Segment area
Perimeter P	`.P`	Boundary length (arc + chord)
Arc length b	`.b`	Segment arc length
Chord s	`.s`	Chord length
Centroid S	`.S`	Distance of centroid from circle center on angle bisector
Radius r	`.r`	Circle radius

Examples

segm(r=5, alpha=60)=

segm(r=5, s=4)=

segm(h=2, s=6)=

Sphere Solver – `sphere`

Syntax

sphere(...=...)

Description

Computes sphere properties from exactly one named input. Supported input names are r (radius), d (diameter), P (great-circle perimeter), S (surface area), A (great-circle cross-section area), and V (volume).

Result List

Label	Key	Description
Radius r	`.r`	Sphere radius
Diameter d	`.d`	Sphere diameter
Perimeter P	`.P`	Great-circle perimeter
Surface area S	`.S`	Sphere surface area
Cross-section area A	`.A`	Great-circle area
Volume V	`.V`	Sphere volume

Examples

sphere(r=5)=

sphere(d=10)=

sphere(P=31.416)=

sphere(S=314.159)=

sphere(A=78.54)=

sphere(V=523.599)=

Bicylinder Solver – `bicylinder`

Syntax

bicylinder(...=...)

Description

Computes properties of a bicylinder (Steinmetz solid) – the intersection of two identical right circular cylinders of equal radius r whose axes are perpendicular and intersect at their centres. Accepts exactly one named input: r, V, or S.

Formulas

Quantity	Formula
Volume V	V = (16/3) r³
Surface area S	S = 16 r²

Result List

Label	Key	Description
Radius r	`.r`	Cylinder radius
Volume V	`.V`	Volume of the intersection solid
Surface S	`.S`	Total surface area

Examples

bicylinder(r=5)=
bicylinder(V=666.667)=
bicylinder(S=400)=

Tricylinder Solver – `tricylinder`

Syntax

tricylinder(...=...)

Description

Computes properties of a tricylinder (Steinmetz solid, three cylinders) – the intersection of three identical right circular cylinders of equal radius r whose axes are mutually perpendicular and intersect at a common centre. Accepts exactly one named input: r, V, or S.

Formulas

Quantity	Formula
Constant k	k = 2 − √2 ≈ 0.5858
Volume V	V = 8k r³
Surface area S	S = 24k r²

Result List

Label	Key	Description
Radius r	`.r`	Cylinder radius
Volume V	`.V`	Volume of the intersection solid
Surface S	`.S`	Total surface area

Examples

tricylinder(r=5)=
tricylinder(V=234.315)=
tricylinder(S=351.472)=

Oloid Solver – `oloid`

Syntax

oloid(...=...)

Description

Computes oloid properties from exactly one named input. Supported input names are r (radius), a (edge length), l (overall length), h (height), S (surface area), and V (volume).

Formulas

Quantity	Formula
Edge length a	a = (4π/3) · r
Length l	l = 3r
Height h	h = 2r
Surface area S	S = 4πr²
Volume V	V ≈ 3.0524184682 · r³

Result List

Label	Key	Description
Radius r	`.r`	Generating circle radius
Edge length a	`.a`	Oloid edge length
Length l	`.l`	Overall oloid length
Height h	`.h`	Overall oloid height
Surface area S	`.S`	Total surface area
Volume V	`.V`	Total volume

Examples

oloid(r=10)=   → a≈41.89, l=30, h=20
oloid(a=41.8879)=
oloid(S=1256.637)=
oloid(V=3052.418)=

Spherical Corner Solver – `sphcor`

Syntax

sphcor(...=...)

Description

Computes spherical-corner (sphere octant) parameters from exactly one named input. Supported input names are r (radius), P (edge arc length), S (octant curved surface area), and V (octant volume).

Result List

Label	Key	Description
Radius r	`.r`	Sphere radius
Arc length P	`.P`	Edge arc length of the spherical corner
Surface area S	`.S`	Curved surface area of the spherical corner
Volume V	`.V`	Volume of the spherical corner

Examples

sphcor(r=5)=

sphcor(P=7.854)=

sphcor(S=39.27)=

sphcor(V=65.45)=

Spherical Shell Solver – `sphsl`

Syntax

sphsl(ri, ro)

Description

Computes properties of a hollow sphere (spherical shell) with constant wall thickness from inner radius ri and outer radius ro. The constraint is ro > ri > 0.

Result List

Label	Key	Description
Shell Volume	`.V`	Volume of shell material
Shell Surface	`.S`	Total shell surface (inner + outer)

Formulas

Quantity	Formula
Shell Volume	V = 4/3 · π · (ro³ − ri³)
Shell Surface	S = 4 · π · (ro² + ri²)

Examples

sphsl(4, 5)=

Spherical Cap Solver – `sphecap`

Syntax

sphecap(r, ...=...)

Description

Computes spherical-cap dimensions from sphere radius r and one named cap input. Supported named inputs are h (cap height) or a (cap base radius).

Result List

Label	Key	Description
Cap height h	`.h`	Cap height
Cap radius a	`.a`	Cap base radius
Sphere radius r	`.r`	Input sphere radius
Cap volume Vs	`.Vs`	Spherical-cap volume
Base area A	`.A`	Area of cap base circle
Cap surface Sc	`.Sc`	Curved cap surface area
Total surface S	`.S`	Cap total surface (Sc + A)

Examples

sphecap(5,h=2)=

sphecap(5,a=3)=

Spherical Sector Solver – `sphesec`

Syntax

sphesec(r, ...=...)

Description

Computes spherical-sector dimensions from sphere radius r and one named segment parameter: sh (segment height) or sr (segment radius).

Result List

Label	Key	Description
Segment height	`.sh`	Height of spherical segment/cap
Segment radius	`.sr`	Radius of segment base circle
Sphere radius	`.r`	Sphere radius
Sector volume	`.V`	Volume of spherical sector
Cone surface	`.Ac`	Lateral surface of cone part
Cap surface	`.As`	Curved spherical cap surface
Sector surface	`.A`	Total sector surface (cone + cap)

Examples

sphesec(5,sh=2)=

sphesec(5,sr=3)=

Spherical Segment Solver – `sphseg`

Syntax

sphseg(r1, r2, h)

Description

Computes parameters of a spherical segment (spherical zone) from two base radii r1, r2 and segment height h.

Result List

Label	Key	Description
Sphere radius	`.r`	Radius of the parent sphere
Lateral surface	`.Al`	Curved lateral area of the spherical segment
Total surface	`.At`	Lateral area plus both base-circle areas
Volume	`.V`	Volume of the spherical segment
Distance to top	`.dt`	Distance from top pole to top base plane
Distance to center	`.dc`	Distance from sphere center to top base plane

Example

sphseg(3, 4, 5)=

Spherical Wedge Solver – `sphwed`

Syntax

sphwed(r, deg)

Description

Computes spherical-wedge parameters from sphere radius r and wedge angle deg. The angle follows the MathBox toolbar DEG/RAD mode.

Result List

Label	Key	Description
Volume V	`.V`	Spherical wedge volume
Outer surface O	`.O`	Outer spherical surface part
Wedge surface W	`.W`	Two planar semicircle cut surfaces
Total surface S	`.S`	Total wedge surface (O + W)

Example

sphwed(5, 60)=

Spherical Ring Solver – `sphring`

Syntax

sphring(rs, rc)

Description

Computes spherical-ring (napkin-ring) parameters for a sphere with radius rs drilled by a centered cylinder with radius rc. The constraint is rs > rc > 0.

Result List

Label	Key	Description
Ring volume VR	`.VR`	Remaining volume of the spherical ring
Cylinder volume VC	`.VC`	Reference cylinder volume with radius rc and height L
Cylinder height L	`.L`	Hole height through the sphere
Ring surface S	`.S`	Total ring surface (outer spherical zone + inner cylindrical surface)

Example

sphring(5, 3)=

Spheroid Solver – `spheroid`

Syntax

spheroid(a, b)

Description

Computes volume and surface area of a spheroid (ellipsoid of revolution) from equatorial semi-axis a and polar semi-axis b. Works for oblate (a > b), prolate (b > a), and sphere (a = b) cases.

Result List

Label	Key	Description
Volume V	`.V`	Spheroid volume
Surface area S	`.S`	Total spheroid surface area

Examples

spheroid(5, 3)=
spheroid(3, 5)=

Triaxial Ellipsoid Solver – `triellip`

Syntax

triellip(a, b, c)

Description

Computes volume and surface area of a triaxial ellipsoid from the semi-axes a, b, and c. The surface area uses the Knud Thomsen approximation.

Result List

Label	Key	Description
Volume V	`.V`	Triaxial ellipsoid volume
Surface area S	`.S`	Triaxial ellipsoid surface area (approx.)

Example

triellip(5, 4, 3)=

Torus Solver – `torus`

Syntax

torus(R=..., r=...)
torus(D=..., d=...)

Description

Computes torus parameters from two named inputs: one major axis (R or D) and one minor axis (r or d). Named arguments are case-sensitive.

R – major radius (center of tube to center of torus)
D – outer diameter of the torus = 2 × (R + r)
r – minor radius (tube radius)
d – inner hole diameter = 2 × (R − r)

Result List

Label	Key	Description
Radius R	`.R`	Major radius (center to tube center)
Radius r	`.r`	Minor radius (tube radius)
Diameter D	`.D`	Outer diameter = 2 × (R + r)
Diameter d	`.d`	Inner hole diameter = 2 × (R − r)
Surface area S	`.S`	Torus surface area
Volume V	`.V`	Torus volume

Examples

torus(R=5, r=2)=      → R=5, r=2, D=14, d=6
torus(D=14, d=6)=     → same result: R=5, r=2

Spindle Torus Solver – `spitor`

Syntax

spitor(R, r)
spitorus(R, r)

Description

Computes parameters of a spindle torus (also called an apple torus). A spindle torus is formed when the tube radius r exceeds the major radius R, causing the surface to self-intersect along the axis and enclose an inner void. Condition: r > R.

R – major radius: distance from the center of the torus to the center of the tube
r – tube radius (must be greater than R)

Result List

Label	Key	Description
Inner height h	`.h`	Distance between the two self-intersection points on the axis = 2 √(r²−R²)
Width b	`.b`	Outer diameter = 2 × (R + r)
Volume V	`.V`	Enclosed volume (exact closed form)

Examples

spitor(2, 5)=
spitorus(2, 5)=

Paraboloid Solver – `paraboloid`

Syntax

paraboloid(s, R)

Description

Computes geometric properties of a paraboloid of revolution defined by the surface equation z = s · ρ², where s is the shape parameter (steepness) and R is the base radius.

s – shape parameter: larger values give a steeper, taller paraboloid
R – radius of the circular base

Formulas

Quantity	Formula
Height h	h = s · R²
Lateral surface L	L = π / (6s²) · [(1 + 4s²R²)^3/2 − 1]
Surface area S	S = L + π R² (lateral + circular base)
Volume V	V = π R² h / 2

Result List

Label	Key	Description
Height h	`.h`	Paraboloid height = s · R²
Lateral surface L	`.L`	Curved side surface (exact closed form)
Surface area S	`.S`	Total surface = lateral + base area
Volume V	`.V`	Volume = π R² h / 2

Examples

paraboloid(0.5, 4)=
paraboloid(1, 3)=

Cone Solver – `cone`

Syntax

cone(r, h)

Description

Computes geometric properties of a right circular cone from base radius r and height h.

Formulas

Quantity	Formula
Base perimeter P	P = 2πr
Slant length L	L = √(r² + h²)
Base area A	A = πr²
Lateral surface SL	SL = πrL
Total surface S	S = A + SL
Volume V	V = πr²h / 3

Result List

Label	Key	Description
Base Perimeter P	`.P`	Perimeter of the circular base
Slant Length L	`.L`	Side edge length from base rim to apex
Base Area A	`.A`	Area of the circular base
Lateral Surface SL	`.SL`	Curved side surface
Total Surface S	`.S`	Base + lateral surface
Volume V	`.V`	Cone volume

Examples

cone(3, 4)=
cone(5, 12)=

Cylinder Solver – `cylinder`

Syntax

cylinder(r, h)

Description

Computes geometric properties of a right circular cylinder from radius r and height h.

Formulas

Quantity	Formula
Volume V	V = πr²h
Surface area S	S = 2πrh + 2πr²
Lateral area L	L = 2πrh
Base area A	A = πr²
Perimeter P	P = 2πr

Result List

Label	Key	Description
Volume V	`.V`	Cylinder volume
Surface Area S	`.S`	Total cylinder surface area
Lateral Area L	`.L`	Side surface area
Base Area A	`.A`	Area of one circular base
Perimeter P	`.P`	Circumference of base circle

Examples

cylinder(3, 10)=
cyl(3, 10)=

Elliptic Cylinder Solver – `ellcyl`

Syntax

ellcyl(a, b, h)
ellcylinder(a, b, h)

Description

Computes geometric properties of a right elliptic cylinder from ellipse semi-axes a, b, and height h.

Formulas

Quantity	Formula
Volume V	V = πabh
Surface area S	S = Ph + 2πab
Lateral area L	L = Ph
Base area A	A = πab
Perimeter P	P ≈ π(3(a+b) - √((3a+b)(a+3b)))

Result List

Label	Key	Description
Volume V	`.V`	Elliptic cylinder volume
Surface Area S	`.S`	Total surface area
Lateral Area L	`.L`	Curved side area
Base Area A	`.A`	Area of one elliptic base
Perimeter P	`.P`	Ellipse base perimeter (Ramanujan approximation)

Examples

ellcyl(5, 3, 10)=
ellcylinder(5, 3, 10)=

Oblique Cylinder Solver – `oblcyl`

Syntax

oblcyl(r, alpha, h)
obliquecyl(r, alpha, h)

Description

Computes geometric properties of an oblique circular cylinder from radius r, inclination angle alpha (to the cylinder axis), and perpendicular height h. The angle follows the toolbar DEG/RAD mode.

Formulas

Quantity	Formula
Height h	h = h (input)
Side length a	a = h / cos(α)
Volume V	V = πr²h
Perimeter P	P = 2πr
Base area A	A = πr²
Lateral area L	L = P·a
Surface area S	S = L + 2A

Result List

Label	Key	Description
Height h	`.h`	Perpendicular cylinder height
Side Length a	`.a`	Generator (side edge) length
Volume V	`.V`	Cylinder volume
Perimeter P	`.P`	Circumference of the circular base
Base Area A	`.A`	Area of one circular base
Lateral Area L	`.L`	Side surface area
Surface Area S	`.S`	Total surface area

Examples

oblcyl(3, 20, 10)=
obliquecyl(3, 20, 10)=

Half Cylinder Solver – `halfcyl`

Syntax

halfcyl(r, h)
semicyl(r, h)

Description

Computes geometric properties of a half cylinder (semi-cylinder) from radius r and height/length h.

Formulas

Quantity	Formula
Volume V	V = πr²h / 2
Lateral area L	L = πrh + 2rh
Surface area S	S = L + πr²
Diagonal d	d = √((2r)² + h²)

Result List

Label	Key	Description
Volume V	`.V`	Half-cylinder volume
Surface Area S	`.S`	Total surface area
Lateral Area L	`.L`	Curved half mantle + rectangular cut face
Diagonal d	`.d`	Diagonal across rectangle 2r × h

Examples

halfcyl(3, 10)=
semicyl(3, 10)=

Oblique Cylinder Section Solver – `oblcylsec`

Syntax

oblcylsec(r, hs, hl)
cylsec(r, hs, hl)

Description

Computes geometric properties of an oblique cylinder section from radius r, short height hs, and long height hl. Use hl ≥ hs.

Formulas

Quantity	Formula
Mean height h̄	h̄ = (hs + hl) / 2
Volume V	V = πr²h̄
Lateral area L	L = 2πr h̄
Base area A	A = πr²
Perimeter P	P = 2πr
Semi-axis a	a = r√(1 + ((hl-hs)/(2r))²)
Surface area S	S = L + A + A_top, with A_top = πra

Result List

Label	Key	Description
Volume V	`.V`	Section volume
Surface Area S	`.S`	Total surface area
Lateral Area L	`.L`	Curved side area
Base Area A	`.A`	Area of base circle
Perimeter P	`.P`	Circumference of base circle
Semi-axis a	`.a`	Top ellipse semi-major axis

Examples

oblcylsec(3, 8, 12)=
cylsec(3, 8, 12)=

Cylindrical Sector Solver – `cylsector`

Syntax

cylsector(r, h, alpha)
cylwedge(r, h, alpha)

Description

Computes geometric properties of a cylindrical sector (sector prism / wedge) from radius r, height h, and central angle alpha. The angle follows the toolbar DEG/RAD mode.

Formulas

Quantity	Formula
Wedge length l	l = rα (with α in radians)
Base area AB	AB = (α/2) r²
Side area AS	AS = 2rh
Lateral area L	L = AS + lh
Volume V	V = ABh
Surface area S	S = L + 2AB

Result List

Label	Key	Description
Volume V	`.V`	Sector volume
Base Area AB	`.AB`	Area of one sector base
Side Area AS	`.AS`	Two radial rectangle faces
Lateral Area L	`.L`	Side area total (radial + curved)
Surface Area S	`.S`	Total surface area
Wedge Length l	`.l`	Arc length of wedge base

Examples

cylsector(3, 10, 60)=
cylwedge(3, 10, 60)=

Cylindrical Segment Solver – `cylseg`

Syntax

cylseg(r, h, l)

Description

Computes geometric properties of a cylindrical segment – the solid cut from a cylinder by a plane parallel to its axis – from radius r, height / length h, and segment depth (sagitta) l where 0 < l < 2r.

Formulas

Quantity	Formula
Half-angle α	α = arccos((r−l)/r)
Segment width s (chord)	s = 2√(2rl−l²)
Arc length b	b = 2rα
End area AT	AT = r²α − (r−l)s/2
Volume V	V = AT · h
Lateral area L	L = b · h
Surface area S	S = L + sh + 2AT

Result List

Label	Key	Description
Segment Width s	`.s`	Chord width of the cross-section
Arc Length b	`.b`	Arc length of the curved cross-section
Volume V	`.V`	Segment volume
Surface Area S	`.S`	Total surface area
Lateral Area L	`.L`	Curved lateral area
End Area AT	`.AT`	Circular segment area of one end face

Examples

cylseg(5, 10, 2)=

Regular N-gonal Pyramid Solver – `ngonpyramid`

Syntax

ngonpyramid(a, h, n)

Description

Computes geometric properties of a regular n-gonal pyramid – a pyramid whose base is a regular polygon with n equal sides of length a, and whose apex is directly above the center of the base at height h. Supports any number of sides n ≥ 3: triangular (n=3), square (n=4), pentagonal (n=5), hexagonal (n=6), etc.

Inputs

Parameter	Description
`a`	Base edge length (all sides equal)
`h`	Perpendicular height from base to apex
`n`	Number of base sides (integer ≥ 3)

Formulas

Quantity	Formula
Base apothem ap	ap = a / (2 · tan(π/n))
Base circumradius rc	rc = a / (2 · sin(π/n))
Slant height s	s = √(ap² + h²)
Lateral edge e	e = √(rc² + h²)
Base area A	A = n · a² / (4 · tan(π/n))
Slant area As	As = a · s / 2 (area of one triangular face)
Lateral surface AL	AL = n · As
Surface area S	S = A + AL
Perimeter P	P = n · a
Volume V	V = A · h / 3

Result list

Label	Key	Description
Slant height s	`.s`	Distance from apex to midpoint of a base edge
Edge length e	`.e`	Distance from apex to a base vertex
Base area A	`.A`	Area of the regular n-gon base
Slant area As	`.As`	Area of one triangular lateral face
Lateral surface AL	`.AL`	Total lateral (side) surface area
Surface area S	`.S`	Total surface area (base + lateral)
Perimeter P	`.P`	Perimeter of the base polygon
Volume V	`.V`	Volume of the pyramid

Examples

ngonpyramid(6, 10, 4)=    Square pyramid, a=6, h=10
ngonpyramid(5, 8, 6)=     Hexagonal pyramid, a=5, h=8
ngonpyramid(4, 7, 3)=     Triangular pyramid, a=4, h=7

Regular N-gonal Double Pyramid Solver – `ngonbipyramid`

Syntax

ngonbipyramid(a, h, n)
ngondoublepyramid(a, h, n)   // alias
bipyramid(a, h, n)           // alias
doublepyramid(a, h, n)       // alias

Description

Computes geometric properties of a regular n-gonal double pyramid (bipyramid), formed by two identical regular pyramids joined base-to-base. The shared base is a regular n-gon with edge length a, and each half has height h.

Inputs

Parameter	Description
`a`	Base edge length of the regular n-gon
`h`	Half-height (height of one pyramid half)
`n`	Number of sides (integer ≥ 3)

Formulas

Quantity	Formula
Slant height `s`	`s = √(ap² + h²)`
Edge length `e`	`e = √(rc² + h²)`
Total height `i`	`i = 2h`
One side face area `As`	`As = a · s / 2`
Surface area `S`	`S = 2 · n · As`
Perimeter `P`	`P = n · a`
Volume `V`	`V = 2 · A · h / 3`, with `A = n · a² / (4 · tan(π/n))`

Result List

Label	Key	Description
Slant height s	`.s`	Face slant height of one pyramid half
Edge length e	`.e`	Lateral edge from apex to base vertex
Total height i	`.i`	Distance between both apexes
Side face area As	`.As`	Area of one triangular side face
Surface area S	`.S`	Total outer surface area
Perimeter P	`.P`	Perimeter of the middle regular n-gon
Volume V	`.V`	Total volume of both pyramid halves

Examples

ngonbipyramid(6,8,4)=
ngonbipyramid(5,7,6)=

Regular N-gonal Pyramid Frustum Solver – `ngonfrustum`

Syntax

ngonfrustum(a, b, h, n)
pyramidfrustum(a, b, h, n)   // alias
pyrfrustum(a, b, h, n)       // alias

Description

Computes geometric properties of a regular n-gonal pyramid frustum (truncated regular pyramid) with base edge length a, top edge length b, vertical height h, and side count n. The two polygons are similar and parallel.

Inputs

Parameter	Description
`a`	Base edge length
`b`	Top edge length
`h`	Perpendicular height
`n`	Number of sides (integer ≥ 3)

Constraint: b ≤ a

Formulas

Quantity	Formula
Base apothem `ap_base`	`ap_base = a / (2 · tan(π/n))`
Top apothem `ap_top`	`ap_top = b / (2 · tan(π/n))`
Slant height `s`	`s = √((ap_base-ap_top)² + h²)`
Lateral edge `e`	`e = √((rc_base-rc_top)² + h²)`
One side face area `As`	`As = (a+b) · s / 2`
Lateral area `Am`	`Am = n · As`
Surface area `S`	`S = A_base + A_top + Am`
Perimeter `P`	`P = n · a`
Volume `V`	`V = h(A_base+A_top+√(A_base · A_top))/3`

Result List

Label	Key	Description
Slant height s	`.s`	Face slant height between base and top polygons
Edge length e	`.e`	Lateral edge from base vertex to top vertex
Side face area As	`.As`	Area of one trapezoid side face
Lateral area Am	`.Am`	Total lateral area of all side faces
Surface area S	`.S`	Total area (base + top + lateral)
Perimeter P	`.P`	Perimeter of the base polygon
Volume V	`.V`	Frustum volume

Examples

ngonfrustum(8,4,6,4)=
ngonfrustum(6,3,9,5)=

Hollow Cylinder Solver – `hollowcyl`

Syntax

hollowcyl(ro, ri, h)

Description

Computes geometric properties of a hollow cylinder (tube) from outer radius ro, inner radius ri, and height h.

Formulas

Quantity	Formula
Volume V	V = π(ro² − ri²)h
Surface area S	S = 2πh(ro + ri) + 2π(ro² − ri²)
Lateral areas L	L = 2πh(ro + ri)
Outer surface Lo	Lo = 2πroh
Inner surface Li	Li = 2πrih
Base area A	A = π(ro² − ri²)
Perimeter P	P = 2π(ro + ri)

Result List

Label	Key	Description
Volume V	`.V`	Hollow cylinder volume
Surface Area S	`.S`	Total outer+inner+base-ring area
Lateral Areas L	`.L`	Outer + inner lateral areas
Outer Surface Lo	`.Lo`	Outer curved side area
Inner Surface Li	`.Li`	Inner curved side area
Base Area A	`.A`	One annular base area
Perimeter P	`.P`	Sum of outer and inner base circumferences

Examples

hollowcyl(6, 4, 10)=
tube(6, 4, 10)=

Frustum Solver – `frustum`

Syntax

frustum(rs, rl, h)

Description

Computes geometric properties of a right circular truncated cone (frustum) from small/top radius rs, large/base radius rl, and height h.

Formulas

Quantity	Formula
Volume V	V = πh(rl² + rl · rs + rs²) / 3
Lateral length L	L = √((rl - rs)² + h²)
Top surface AT	AT = πrs²
Base surface AB	AB = πrl²
Lateral surface SL	SL = π(rl + rs)L
Total surface S	S = AT + AB + SL
Large perimeter PL	PL = 2πrl
Small perimeter PS	PS = 2πrs

Result List

Label	Key	Description
Volume V	`.V`	Frustum volume
Lateral Length L	`.L`	Slant side length
Top Surface AT	`.AT`	Area of the top circle
Base Surface AB	`.AB`	Area of the base circle
Lateral Surface SL	`.SL`	Curved side area
Total Surface S	`.S`	Total area = top + base + lateral
Large Perimeter PL	`.PL`	Circumference of base circle
Small Perimeter PS	`.PS`	Circumference of top circle

Examples

frustum(2, 5, 6)=
frustum(3, 8, 10)=

Double Cone Solver – `bicone`

Syntax

bicone(r, h)

Description

Computes geometric properties of a double cone (bicone) made of two identical right circular cones joined at the base, with base radius r and cone height h (per cone).

Formulas

Quantity	Formula
Base area A	A = πr²
Lateral areas SL	SL = 2πr√(r² + h²)
Total surface S	S = SL
Volume V	V = 2πr²h / 3

Result List

Label	Key	Description
Base Area A	`.A`	Shared base circle area
Lateral Areas SL	`.SL`	Sum of both lateral cone surfaces
Total Surface S	`.S`	Total outer area (equals SL)
Volume V	`.V`	Sum of both cone volumes

Examples

bicone(3, 4)=
doublecone(3, 4)=

Pointed Pillar Solver – `pillar`

Syntax

pillar(a, b, hp, hc)

Description

Computes geometric properties of a pointed pillar composed of a truncated cone and a top cone. Inputs are base radius a, transition radius b, pillar base height hp, and cone height hc.

Formulas

Quantity	Formula
Base area A	A = πa²
Lateral area L	L = π(a+b)√((a-b)² + hp²) + πb√(b² + hc²)
Surface area S	S = A + L
Volume V	V = πhp(a² + ab + b²)/3 + πb²hc/3

Result List

Label	Key	Description
Base Area A	`.A`	Bottom circular base area
Lateral Area L	`.L`	Frustum lateral area + cone lateral area
Surface Area S	`.S`	Total outer area = A + L
Volume V	`.V`	Frustum volume + cone volume

Examples

pillar(8, 5, 10, 6)=
spitzpfeiler(8, 5, 10, 6)=

Rounded Cone Solver – `spherocone`

Syntax

spherocone(R, r, hf)

Description

Computes geometric properties of a rounded cone made of a frustum and a spherical cap, with tangent-continuous transition. Inputs are base radius R, cap sphere radius r, and frustum height hf.

Formulas

Quantity	Formula
Transition radius x	hf√(r²−x²) = x(R−x), solved numerically
Cap height i	i = r − √(r²−x²)
Total height h	h = hf + i
Surface area S	S = πR² + π(R+x)√((R−x)²+hf²) + 2πri
Volume V	V = πhf(R²+Rx+x²)/3 + πi²(3r−i)/3
Base angle α	α = atan(hf / (R−x))

Result List

Label	Key	Description
Cap Height i	`.i`	Spherical cap height
Total Height h	`.h`	Overall shape height
Surface Area S	`.S`	Total outer surface area
Volume V	`.V`	Total volume (frustum + cap)
Base Angle α	`.alpha`	Frustum base angle in degrees

Examples

spherocone(8, 5, 10)=
roundcone(8, 5, 10)=

Elliptic Frustum Solver – `ellfrustum`

Syntax

ellfrustum(a, b, h, ht)

Description

Computes geometric properties of a right truncated elliptic cone from bottom semi-axes a, b, full cone height h, and truncated top-cone height ht measured from the apex.

Formulas

Quantity	Formula
Top semi-axis c	c = a · ht / h
Top semi-axis d	d = b · ht / h
Volume V	V = π(h-ht)(ab + cd + √(ab · cd)) / 3
Base area A	A = πab
Lateral surface M	M = M_{full cone} · (1-(ht/h)²)
Total surface S	S = A + (πcd) + M

Result List

Label	Key	Description
Top Semi-axis c	`.c`	Top ellipse semi-axis corresponding to a
Top Semi-axis d	`.d`	Top ellipse semi-axis corresponding to b
Volume V	`.V`	Elliptic frustum volume
Base Area A	`.A`	Bottom ellipse area (πab)
Lateral Surface M	`.M`	Curved side area
Total Surface S	`.S`	Bottom base + top base + lateral surface

Examples

ellfrustum(8, 5, 12, 4)=
ellfrustum(10, 6, 15, 3)=

Elliptic Cone Solver – `ellcone`

Syntax

ellcone(a, b, h)

Description

Computes geometric properties of a right elliptic cone from major and minor base semi-axes a, b, and height h.

Formulas

Quantity	Formula
Base area A	A = πab
Volume V	V = πab h / 3
Lateral area M	Computed numerically from the exact elliptic-cone surface integral
Total surface S	S = A + M

Result List

Label	Key	Description
Volume V	`.V`	Elliptic cone volume
Base Area A	`.A`	Area of elliptical base
Lateral Area M	`.M`	Curved side area
Total Surface S	`.S`	Base + lateral area

Examples

ellcone(4, 2, 6)=
ellcone(10, 6, 12)=

Solid Angle Solver – `solang`

Syntax

solang(W=..., R=...)
        solang(W=..., A=...)
        solang(A=..., R=...)

Description

Solves the solid angle relation Ω = A / R². Provide any two of the three quantities; the third is computed. Parameters are case-sensitive.

Parameter	Symbol	Description
`W`	Ω	Solid angle [sr – steradian]. Aliases: `Ω`, `omega`
`A`	A	Spherical cap area intercepted on a sphere of radius R
`R`	R	Sphere radius

Formulas

Unknown	Formula
W (solid angle)	Ω = A / R²
A (cap area)	A = Ω · R²
R (sphere radius)	R = √(A / Ω)

Note: a full sphere subtends 4π ≈ 12.566 sr.

Examples

solang(A=12.566, R=2)    → W ≈ 3.1416 sr
        solang(W=3.1416, R=2)   → A ≈ 12.566
        solang(W=3.1416, A=12.566) → R = 2

Ellipse Solver – `ellipse`

Syntax

ellipse(a, b)

Description

Computes the perimeter and area of an ellipse from the two semi-axes a and b.

Formulas

Quantity	Symbol	Formula	Unit
Perimeter	`P`	π · (3(a+b) − √((3a+b)(a+3b)))	–
Area	`A`	πab	–

Result List

Label	Key	Description
Semi-axis a	`.a`	Input semi-axis a
Semi-axis b	`.b`	Input semi-axis b
Perimeter P	`.P`	Ellipse perimeter (Ramanujan approximation)
Area A	`.A`	Ellipse area

Examples

ellipse(5, 3)=

Semi-axis a    5
Semi-axis b    3
Perimeter P 25.527
Area A         47.124

Circular Arc (angle input) – `arc`

Syntax

arc(r, alpha)

Description

Computes all characteristic dimensions of a circular arc (Kreisbogen / Rundbogen) given the radius r and the central angle α in degrees.

Formulas

Quantity	Symbol	Formula	Unit
Arc length	`s`	r · α_rad	–
Chord	`c`	2r · sin(α/2)	–
Sagitta (Pfeilhöhe)	`h`	r · (1 − cos(α/2))	–
Sector area	`A`	½ r² · α_rad	–
Segment area	`As`	A − ½ r² · sin(α)	–

Result List

Label	Key	Description
Radius	`.r`	Input radius
Angle α	`.alpha`	Central angle in degrees (input)
Arc length	`.s`	Length of the arc
Chord	`.c`	Straight line connecting the two arc endpoints
Sagitta	`.h`	Height from chord midpoint to arc midpoint (Pfeilhöhe)
Sector area	`.A`	Area of the pie-slice sector
Segment area	`.As`	Area of the circular segment (sector minus triangle)

Examples

arc(5, 90)=

Radius          5
Angle α         90 °
Arc length      7.854
Chord           7.071
Sagitta         1.464
Sector area     19.635
Segment area    7.135

arc(10, 180)=

Arc length      31.416 (half-circle)
Chord           20
Sagitta         10
Sector area     157.08
Segment area    157.08 (= half disk)

r := arc(5, 90)
r.s      → 7.854
r.c      → 7.071
r.h      → 1.464
r.A      → 19.635

Circular Arc (arc-length input) – `arcl`

Syntax

arcl(r, s)

Description

Computes all characteristic dimensions of a circular arc given the radius r and the arc length s. The central angle is derived as α = s / r [rad]. All other outputs are identical to arc.

Examples

arcl(5, 7.854)=

Radius          5
Angle α         90 °
Arc length      7.854
Chord           7.071
Sagitta         1.464
Sector area     19.635
Segment area    7.135

r := arcl(5, 7.854)
r.alpha  → 90°
r.c      → 7.071

Rhombus Solver – `rhom`

Syntax

rhom(a, h)
rhombus(a, h)   // alias

Description

Computes a rhombus from side length a and height h. The function returns the geometric base set: area, perimeter, both diagonals, and both interior angles.

Formulas

Quantity	Formula	Unit
`A` – area	a × h	–
`P` – perimeter	4a	–
`alpha` – acute angle	asin(h/a) × 180/π	°
`beta` – obtuse angle	180° − alpha	°

Result List

Label	Key	Description
Side length a	`.a`	Input side length
Height h	`.h`	Input height
Area A	`.A`	Rhombus area
Perimeter P	`.P`	Total edge length
Diagonal e	`.e`	Longer diagonal
Diagonal f	`.f`	Shorter diagonal
Angle α	`.alpha`	Acute interior angle
Angle β	`.beta`	Obtuse interior angle

Examples

rhom(10, 6)=

Side length a    10
Height h        6
Area A          60
Perimeter P      40
Diagonal e       15.211
Diagonal f       7.889
Angle α        36.87 °
Angle β        143.13 °

r := rhom(12, 8)
r.A      → 96
r.P      → 48
r.alpha  → 41.81

Rhombus from Diagonals – `rhomd`

Syntax

rhomd(e, f)
rhombusdf(e, f)   // alias

Description

Computes a rhombus from its two diagonals e and f. The function returns area, perimeter, and both interior angles.

Formulas

Quantity	Formula	Unit
`A` – area	e × f / 2	–
`a` – side length	½ √(e² + f²)	–
`P` – perimeter	4a = 2 √(e² + f²)	–
`alpha` – acute angle	2 × atan(min(e,f) / max(e,f)) × 180/π	°
`beta` – obtuse angle	180° − alpha	°

Result List

Label	Key	Description
Area A	`.A`	Rhombus area
Perimeter P	`.P`	Total edge length
Angle α	`.alpha`	Acute interior angle
Angle β	`.beta`	Obtuse interior angle

Examples

rhomd(16, 12)=

Area A          96
Perimeter P      40
Angle α        73.74 °
Angle β        106.26 °

r := rhomd(20, 12)
r.A      → 120
r.P      → 46.648
r.alpha  → 61.93

Kite Area – `kitearea`

Syntax

kitearea(e, f)
deltoidarea(e, f)   // alias

Description

Computes the area of a kite (deltoid) directly from its two diagonals.

Formula

Quantity	Formula	Unit
`A` – area	e × f / 2	–

Examples

kitearea(10, 6)   → 30
deltoidarea(12, 8) → 48

Kite (Deltoid) Solver – `kite`

Syntax

kite(e, f, c)
deltoid(e, f, c)   // alias

Description

Computes a kite (deltoid) from its diagonals and split position on the main diagonal. Inputs are: e (full main diagonal), f (secondary diagonal), and c (distance from one endpoint of diagonal e to the intersection point of both diagonals).

Formulas

Quantity	Formula	Unit
`a` – first side pair	√(c² + (f/2)²)	–
`b` – second side pair	√((e-c)² + (f/2)²)	–
`P` – perimeter	2(a + b)	–
`A` – area	e × f / 2	–
`alpha` – tip angle at c-side	2 × atan((f/2)/c) × 180/π	°
`beta` – opposite tip angle	2 × atan((f/2)/(e-c)) × 180/π	°
`gamma` – side angles	(360° − alpha − beta)/2	°

Result List

Label	Key	Description
Side a	`.a`	First pair of equal sides
Side b	`.b`	Second pair of equal sides
Perimeter	`.P`	Total boundary length
Area A	`.A`	Kite area
Angle α	`.alpha`	First tip angle
Angle β	`.beta`	Opposite tip angle
Angle γ	`.gamma`	Both side angles (equal)

Examples

kite(10, 6, 4)=

Side a          5
Side b          6.708
Perimeter       23.416
Area A          30
Angle α        73.74 °
Angle β        53.13 °
Angle γ      116.565 °

k := deltoid(12, 8, 5)
k.a      → 6.403
k.b      → 8.062
k.A      → 48
k.P      → 28.93

Half-Square Kite Solver – `kitehs`

Syntax

kitehs(a, b)

Description

Computes a half-square kite from two side lengths. The upper tip angle is fixed to 90° (isosceles right top part).

Constraint

b > a/√2

Results

Label	Key	Description
Perimeter P	`.P`	Total boundary length
Area A	`.A`	Area
Diagonal e	`.e`	Main diagonal
Section e₁	`.e1`	Upper part of diagonal e
Section e₂	`.e2`	Lower part of diagonal e
Diagonal f	`.f`	Secondary diagonal
Angle β	`.beta`	Bottom tip angle
Angle γ	`.gamma`	Side angles (equal)
Inner radius ri	`.ri`	Inradius of tangential kite

Example

kitehs(5, 7)=

Right Kite Solver – `kitert`

Syntax

kitert(a, b)

Description

Computes a right-angled kite (right deltoid) from two side lengths. The top angle is fixed to α = 90°.

Constraint

b > a/√2

Results

Label	Key	Description
Perimeter P	`.P`	Total boundary length
Area A	`.A`	Area
Diagonal e	`.e`	Main diagonal
Diagonal f	`.f`	Secondary diagonal
Angle α	`.alpha`	Top angle (fixed at 90°)
Angle γ	`.gamma`	Equal side angles
Incircle radius ri	`.ri`	Inscribed-circle radius
Outer radius rc	`.rc`	Circumradius of the top right triangle

Example

kitert(3, 5)=

Arbitrary Quadrilateral Solver – `quad`

Syntax

quad(a, b, c, beta, gamma)
quadrilateral(a, b, c, beta, gamma)   // alias

Description

Computes a general convex quadrilateral from three consecutive sides a, b, c and two interior angles beta and gamma. The angles follow the toolbar angle mode (DEG/RAD).

Naming Scheme (Sides, Angles, Diagonals)

            B (β)
           / \
        a /   \ b
         /  f  \
     A (α)---e---C (γ)
         \       /
       d  \     / c
           \   /
            D (δ)

Sides:     a = AB, b = BC, c = CD, d = DA
Diagonals: e = AC, f = BD
Angles:    α at A, β at B, γ at C, δ at D

Input Constraints

a,b,c > 0, 0 < beta < 180°, 0 < gamma < 180°, and the resulting shape must be convex.

Result List

Label	Key	Description
Side length d	`.d`	Fourth side
Diagonal e	`.e`	Diagonal AC
Diagonal f	`.f`	Diagonal BD
Area A	`.A`	Quadrilateral area
Perimeter U	`.U`	Total boundary length
Angle α	`.alpha`	Interior angle at A
Angle δ	`.delta`	Interior angle at D

Example

quad(5, 4, 6, 110, 95)=

Concave Quadrilateral Solver – `quadconc`

Syntax

quadconc(a, b, c, alpha, gamma)
concquad(a, b, c, alpha, gamma)   // alias

Description

Computes a general concave quadrilateral from three consecutive sides a, b, c and two interior angles alpha and gamma. The angles follow the toolbar angle mode (DEG/RAD).

Naming Scheme (Sides, Angles, Diagonals)

            B (β)
           / \
        a /   \ b
         /  f  \
     A (α)---e---C (γ)
          \     /
           \   / c
            \ /
             D (δ)

Concave case: α is the reflex interior angle at A (> 180°).
Sides:     a = AB, b = BC, c = CD, d = DA
Diagonals: e = AC, f = BD
Angles:    α at A, β at B, γ at C, δ at D

Input Constraints

a,b,c > 0, 180° < alpha < 360°, 0 < gamma < 180°.

Result List

Label	Key	Description
Side d	`.d`	Fourth side
Diagonal e	`.e`	Diagonal AC
Diagonal f	`.f`	Diagonal BD
Area A	`.A`	Quadrilateral area
Perimeter U	`.U`	Total boundary length
Angle α	`.alpha`	Given concave interior angle
Angle δ	`.delta`	Fourth interior angle

Example

quadconc(5, 4, 6, 220, 70)=

Three-Equal-Sides Trapezoid Solver – `treptri`

Syntax

treptri(a, b)

Description

Computes a trapezoid with one distinct base side a and three equal sides of length b (top base and both legs). The function returns geometric parameters including height, diagonal, angles, area, and circumradius.

Naming Scheme

  D (β)────────────b────────────C (β)
   /                               \
  b                                 b
   \                               /
    A (α)────────────a────────────B (α)

Given:   a = bottom base, b = top base and both legs
Output:  d = equal diagonal length, h = height, m = middle width,
         A = area, P = perimeter, rc = circumradius

Formulas

Quantity	Formula
Horizontal shift x	(a - b) / 2
Height h	√(b² - x²)
Diagonal d	√(((a+b)/2)² + h²)
Middle width m	(a + b) / 2
Area A	m × h
Perimeter P	a + 3b
Angle α	atan(h / x) × 180/π
Angle β	180° - α
Circumradius rc	b × d / (2h)

Input Constraints

a > 0, b > 0, and a < 3b.

Result List

Label	Key	Description
Diagonal d	`.d`	Equal diagonal length
Height h	`.h`	Perpendicular height
Middle width m	`.m`	Mean of both bases
Area A	`.A`	Trapezoid area
Perimeter P	`.P`	Total boundary length
Angle α	`.alpha`	Bottom interior angle
Angle β	`.beta`	Top interior angle
Circumradius rc	`.rc`	Circumcircle radius

Example

treptri(10, 4)=

t := treptri(10, 4)
t.d      → 8.944
t.h      → 2.646
t.A      → 18.520
t.P      → 22
t.alpha  → 41.409
t.beta   → 138.591
t.rc     → 6.761

Symmetric Trapezoid Solver – `traps`

Syntax

traps(a, c, h)

Description

Computes a symmetric trapezoid from the two parallel sides a (bottom), c (top), and the height h. In the symmetric case, both legs are equal (b = d), both diagonals are equal (e = f), and the side overhangs are equal.

Naming Scheme

  D (δ)────────────c────────────C (γ)
   /                               \
 d /                                 \ b
  /                                   \
 A (α)──────────────a──────────────B (β)
   |← x →|                     |← x →|

Parallel sides:  a (bottom), c (top)
Legs:            b = d
Diagonals:       e = f
Angles:          α = β,  γ = δ
Overhang:        x = (a - c) / 2

Formulas

Quantity	Formula
Overhang x	(a - c) / 2
Legs b=d	√(h² + x²)
Diagonal e=f	√((x + c)² + h²)
Midline m	(a + c) / 2
Area A	m × h
Perimeter P	a + c + 2b
Angles α=β	atan(h / x) × 180/π
Angles γ=δ	180° - α

Input Constraints

a ≥ c > 0 and h > 0.

Result List

Label	Key	Description
Side a	`.a`	Bottom parallel side
Legs b=d	`.b`	Equal leg length
Side c	`.c`	Top parallel side
Diagonal e=f	`.e`	Equal diagonals
Height h	`.h`	Perpendicular height
Midline m	`.m`	Mean width
Area A	`.A`	Trapezoid area
Perimeter P	`.P`	Total boundary length
Angles α=β	`.alpha`	Bottom equal angles
Angles γ=δ	`.gamma`	Top equal angles
Overhang x	`.x`	Equal side overhang

Example

traps(10, 6, 4)=

t := traps(10, 6, 4)
t.A       → 32
t.P       → 24.944
t.alpha   → 63.435
t.x       → 2

Right Trapezoid Solver – `trapr`

Syntax

trapr(a, c, b=...)
trapr(a, c, d=...)

Description

Computes a right trapezoid from base side a, top side c, and exactly one named third parameter: b (perpendicular side / height) or d (slanted side). The first two parameters are positional (a, c), the third parameter must be named.

Naming Scheme

 D (δ)────────────c────────────C (90°)
  |                              /
  | b (=height)                / d
  |                          /
 A (90°)────────────a────────B (α)

Sides:     a = AB (bottom), b = AD (perpendicular), c = DC (top), d = BC (slanted)
Diagonals: e = AC, f = BD
Angles:    α at B, δ at D

Formulas

Quantity	Formula
Horizontal offset	\|a - c\|
d from b	√((a-c)² + b²)
b from d	√(d² - (a-c)²)
Diagonal e	√(c² + b²)
Diagonal f	√(a² + b²)
Midline m	(a + c) / 2
Area A	m × b
Perimeter P	a + b + c + d
Angle α	atan(b / \|a-c\|) × 180/π
Angle δ	180° - α

Input Constraints

a,c > 0 and either b > 0 or d > |a-c|.

Result List

Label	Key	Description
Side a	`.a`	Bottom base
Side b	`.b`	Perpendicular side (height)
Side c	`.c`	Top side
Side d	`.d`	Slanted side
Diagonal e	`.e`	Diagonal AC
Diagonal f	`.f`	Diagonal BD
Midline m	`.m`	Mean width
Area A	`.A`	Trapezoid area
Perimeter P	`.P`	Total boundary length
Angle α	`.alpha`	Acute angle at B
Angle δ	`.delta`	Obtuse angle at D

Example

trapr(12, 8, b=5)=
trapr(12, 8, d=6)=

r := trapr(12, 8, b=5)
r.A      → 50
r.P      → 31.403
r.alpha  → 51.340
r.delta  → 128.660

Helix (Schraubenlinie) – `helix`

Syntax

helix(r, h, n)

Description

Computes the characteristic properties of a helix (Schraubenlinie / Wendelkurve) given the cylinder radius r, the total axial height h, and the number of turns n. The function uses the standard differential-geometry parametrization (r cos t, r sin t, bt) where b = pitch / (2π).

Formulas

Quantity	Symbol	Formula	Unit
Pitch (Ganghöhe)	`p`	h / n	–
Helix angle	`alpha`	atan(p / 2πr) × 180/π	°
Arc length	`s`	2πn · √(r² + b²) where b = p/(2π)	–
Curvature	`kappa` (κ)	r / (r² + b²)	–
Torsion	`tau` (τ)	b / (r² + b²)	–

Result List

Label	Key	Description
Pitch	`.p`	Height per full turn (Ganghöhe)
Helix angle α	`.alpha`	Angle between tangent and horizontal plane [°]
Arc length	`.s`	Total length of the helix curve
Curvature κ	`.kappa`	Reciprocal of the osculating-circle radius
Torsion τ	`.tau`	Rate of rotation of the osculating plane

Examples

helix(1, 10, 5)=

Pitch            2
Helix angle α    17.657 °
Arc length       64.659
Curvature κ      0.282
Torsion τ        0.090

helix(0.05, 0.3, 10)=

Pitch            0.03
Helix angle α    5.442 °
Arc length       3.144
Curvature κ      19.919
Torsion τ        1.896

r := helix(1, 10, 5)
r.p      → 2
r.s      → 64.659
r.kappa  → 0.282
r.tau    → 0.090

Euclidean Vector Distance – `distance`

Syntax

distance(a, b)

Description

Returns the Euclidean distance ‖a − b‖ between two scalars or, when a and b are array variables, between two n-dimensional vectors. This is the general-purpose distance function; for 2D coordinate work with labelled results use p2dist instead.

Formulas

Input type	Formula
Scalars	\|a − b\|
Vectors / arrays	√∑(aᵢ − bᵢ)²

Examples

distance(3, 7)      → 4
distance(0, 5)      → 5

u := [1, 2, 3]
v := [4, 6, 3]
distance(u, v)      → 5

Quick Reference

Function	Panel group	Parameters	Primary result
`p2dist(Ax,Ay,Bx,By)`	Lines and Curves	Coordinates of two points A, B	c = total distance A↔B; b, a = x/y components; alpha = angle
`p2linedist(Ax,Ay,Bx,By,Px,Py)`	Lines and Curves	Two line points A, B and query point P	d = shortest distance; Fx, Fy = foot point; \|AB\| = line length
`l2angle(Ax,Ay,Bx,By,Cx,Cy,Dx,Dy)`	Lines and Curves	Two points per line: A,B = line 1; C,D = line 2	α = acute intersection angle [0°…90°]
`v2angle(Ax,Ay,Bx,By)`	Lines and Curves	Components of two 2D vectors A and B	α = angle [0°…180°]; dot, \|A\|, \|B\|
`v3angle(Ax,Ay,Az,Bx,By,Bz)`	Lines and Curves	Components of two 3D vectors A and B	α = angle [0°…180°]; dot, \|A\|, \|B\|
`abcangles(a,b,c)`	Lines and Curves	Triangle side lengths a, b, c	α, β, γ = interior angles opposite a, b, c [°]
`polyangles(n)`	Lines and Curves	Number of sides n (integer, n ≥ 3)	sum = interior angle sum; interior = one interior angle; exterior = one exterior angle [°]
`eqtri(...=...)`	Lines and Curves	One named input: a (side), h (height), u (perimeter), or A (area)	Returns full equilateral triangle set: a, h, u, A
`isotri(...=..., ...=...)`	Triangle	Two named inputs following pair rule: one of (a\|h) and one of (b\|h\|A\|alpha)	Returns a, b, h, P, A, alpha, gamma
`rtri(...=..., ...=...)`	Triangle	One side a\|b\|c plus alpha	Returns a, b, c, h, A, P, p, q, alpha, beta
`pythag(...=..., ...=...)`	Triangle	Two known sides from a\|b\|c	Returns a, b, c, h, A, P, p, q, alpha, beta
`isortri(...=...)`	Triangle	One named input: a (base), b (side), h (height), or A (area)	Returns a, b, h, P, A
`triarea(base,height)`	Triangle	Base, height	A = base × height / 2
`triangarea(a,b,alpha)`	Triangle	Two sides a, b and included angle alpha [°]	A = 1/2 × a × b × sin(alpha)
`triarea2ang(base,alpha,beta)`	Triangle	Base and two angles alpha, beta [° or rad]	A = base² × sin(alpha) × sin(beta) / (2 × sin(alpha+beta))
`triheron(a,b,c)`	Triangle	Three side lengths a, b, c	A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
`square(...=...)`	Quadrilaterals	One named input: seite\|diagonal\|umfang\|flaeche (or a\|d\|U\|A)	Returns a, d, U, A, ri, Ai, ru, Au
`goldrect(...=...)`	Quadrilaterals	One named input: a\|b\|d\|U\|A (aliases: laenge\|breite\|diagonal\|umfang\|flaeche)	Returns a, b, d, U, A, phi
`para(a,b,alpha)`	Quadrilaterals	Sides a, b and included angle alpha [°]	Returns A, P, ha, hb, e, f, beta
`paraaera(base,height)`	Quadrilaterals	Base and height	A = base × height
`rect(...=..., ...=...)`	Quadrilaterals	Two named inputs: one of (l\|b) and one of (b\|d\|A\|U)	Returns l, b, d, U, A
`slope(rise, run)`	Lines and Curves	Vertical rise and horizontal run	m = slope; α [°]; grade [%]; rise; run
`p2slope(Ax,Ay,Bx,By)`	Lines and Curves	Coordinates of two points A, B	m = slope; α [°]; grade [%]; Δy (rise); Δx (run)
`circle(...=...)`	Lines and Curves	One named input: r\|d\|U\|A	Returns radius r, diameter d, perimeter U, area A
`sphere(...=...)`	Spherical Solids	One named input: r\|d\|P\|S\|A\|V	Returns r, d, P, S, A, V
`bicylinder(...=...)`	Cylinders	One named input: r\|V\|S	Returns r, V, S
`tricylinder(...=...)`	Cylinders	One named input: r\|V\|S	Returns r, V, S
`oloid(...=...)`	Spherical Solids	One named input: r\|a\|l\|h\|S\|V	Returns r, a, l, h, S, V
`sphcor(...=...)`	Spherical Solids	One named input: r\|P\|S\|V (sphere octant)	Returns r, P, S, V
`sphsl(ri, ro)`	Spherical Solids	Inner radius ri, outer radius ro	Returns shell volume V and shell surface S
`sphecap(r, x=...)`	Spherical Solids	Sphere radius r and one named input: h\|a	Returns h, a, r, Vs, A, Sc, S
`sphesec(r, x=...)`	Spherical Solids	Sphere radius r and one named input: sh\|sr	Returns sh, sr, r, V, Ac, As, A
`sphseg(r1, r2, h)`	Spherical Solids	Two base radii r1, r2 and height h	Returns r, Al, At, V, dt, dc
`sphwed(r, deg)`	Spherical Solids	Sphere radius r and wedge angle deg	Returns V, O, W, S
`sphring(rs, rc)`	Spherical Solids	Sphere radius rs and cylinder radius rc	Returns VR, VC, L, S
`spheroid(a, b)`	Spherical Solids	Equatorial semi-axis a and polar semi-axis b	Returns V, S
`triellip(a, b, c)`	Spherical Solids	Semi-axes a, b, c	Returns V, S
`torus(...=..., ...=...)`	Spherical Solids	One major axis (R\|D) and one minor axis (r\|d)	Returns R, r, D, d, S, V
`spitor(R, r)`	Spherical Solids	Major radius R and tube radius r, with r > R	Returns inner height h, width b, volume V
`paraboloid(s, R)`	Spherical Solids	Shape parameter s and base radius R	Returns height h, lateral surface L, surface area S, volume V
`cone(r, h)`	Cones	Base radius r and cone height h	Returns P, L, A, SL, S, V
`cylinder(r, h)`	Cylinders	Radius r and height h	Returns V, S, L, A, P
`hollowcyl(ro, ri, h)`	Cylinders	Outer radius ro, inner radius ri, height h	Returns V, S, L, Lo, Li, A, P
`ellcyl(a, b, h)`	Cylinders	Ellipse semi-axes a,b and height h	Returns V, S, L, A, P
`oblcyl(r, alpha, h)`	Cylinders	Radius r, inclination angle alpha, height h	Returns h, a, V, P, A, L, S
`halfcyl(r, h)`	Cylinders	Radius r and height h	Returns V, S, L, d
`oblcylsec(r, hs, hl)`	Cylinders	Radius r, short height hs, long height hl	Returns V, S, L, A, P, a
`cylsector(r, h, alpha)`	Cylinders	Radius r, height h, central angle alpha	Returns V, AB, AS, L, S, l
`cylseg(r, h, l)`	Cylinders	Radius r, height h, segment depth l	Returns s, b, V, S, L, AT
`ngonpyramid(a, h, n)`	Pyramids	Base edge a, height h, number of sides n (≥ 3)	Returns s, e, A, As, AL, S, P, V
`ngonbipyramid(a, h, n)`	Pyramids	Base edge a, half-height h, number of sides n (≥ 3)	Returns s, e, i, As, S, P, V
`ngonfrustum(a, b, h, n)`	Pyramids	Base edge a, top edge b, height h, number of sides n (≥ 3)	Returns s, e, As, Am, S, P, V
`frustum(rs, rl, h)`	Cones	Small radius rs, large radius rl, and height h	Returns V, L, AT, AB, SL, S, PL, PS
`bicone(r, h)`	Cones	Base radius r and cone height h	Returns A, SL, S, V
`pillar(a, b, hp, hc)`	Cones	Base radius a, transition radius b, base height hp, cone height hc	Returns A, L, S, V
`spherocone(R, r, hf)`	Cones	Base radius R, cap sphere radius r, frustum height hf	Returns i, h, S, V, α
`ellfrustum(a, b, h, ht)`	Cones	Bottom semi-axes a,b; full cone height h; truncated height ht	Returns c, d, V, A, M, S
`ellcone(a, b, h)`	Cones	Major/minor semi-axis a,b and height h	Returns V, A, M, S
`solang(...=..., ...=...)`	Spherical Solids	Any two of: W (solid angle [sr]), A (cap area), R (sphere radius)	Returns the missing quantity; Ω = A / R²
`sect(r, x=...)`	Lines and Curves	Radius r and one named input: alpha\|A\|P\|b\|s	Returns α, A, P, b, s, S, h
`segm(...=..., ...=...)`	Lines and Curves	Two named inputs: (r\|h) and (alpha\|s)	Returns α, h, A, P, b, s, S, r
`ellipse(a, b)`	Lines and Curves	Two semi-axes a and b	Returns perimeter P and area A
`arc(r, alpha)`	Lines and Curves	Radius, central angle [°]	s = arc length; c = chord; h = sagitta; A = sector area; As = segment area
`arcl(r, s)`	Lines and Curves	Radius, arc length	α [°]; c = chord; h = sagitta; A = sector area; As = segment area
`rhom(a, h)`	Lines and Curves	Side length and height	a, h, A, P, e, f, α, β
`rhomd(e, f)`	Lines and Curves	Diagonals e and f	A, P, α, β
`kitearea(e, f)`	Lines and Curves	Diagonals e and f	A
`kite(e, f, c)`	Lines and Curves	Main diagonal e, secondary diagonal f, split distance c	a, b, P, A, α, β, γ
`kitehs(a, b)`	Lines and Curves	Two side lengths a and b	P, A, e, e1, e2, f, β, γ, ri
`kitert(a, b)`	Lines and Curves	Two side lengths a and b	P, A, e, f, α, γ, ri, rc
`quad(a, b, c, beta, gamma)`	Lines and Curves	Sides a,b,c and angles β,γ (DEG/RAD mode)	d, e, f, A, U, α, δ
`quadconc(a, b, c, alpha, gamma)`	Lines and Curves	Sides a,b,c and angles α,γ (DEG/RAD mode)	d, e, f, A, U, α, δ
`treptri(a, b)`	Lines and Curves	Base a and three equal sides b	d, h, m, A, P, α, β, rc
`traps(a, c, h)`	Lines and Curves	Parallel sides a,c and height h (symmetric trapezoid)	a, b=d, c, e=f, h, m, A, P, α=β, γ=δ, x
`trapr(a, c, b=...)` / `trapr(a, c, d=...)`	Lines and Curves	Positional a,c and one named third parameter b or d	a, b, c, d, e, f, m, A, P, α, δ
`trap(a, c, h, x=...)` / `trap(a, c, h, β=...)`	Lines and Curves	First three positional (a,c,h), fourth parameter named (x\|y\|α\|β\|γ\|δ)	a, b, c, d, e, f, h, m, A, P, α, β, γ, δ, x, y
`helix(r, h, n)`	Lines and Curves	Radius, total height, number of turns	p = pitch; α = helix angle [°]; s = arc length; κ = curvature; τ = torsion
`distance(a,b)`	General	Two scalars or array vectors	‖a−b‖ as a scalar
`deg2rad(x)`	Angle → Conversion	Angle in degrees	Equivalent in radians
`rad2deg(x)`	Angle → Conversion	Angle in radians	Equivalent in degrees
`dms2deg(d, m, s)`	Angle → Conversion	Degrees, minutes, seconds	Decimal degrees
`angpct(deg=...)` / `angpct(pct=...)`	Angle → Conversion	Named argument: degree or percent grade	Degrees [°] ↔ slope grade [%]
`norm360(x)`	Angle → Normalization	Any angle [°]	Equivalent in [0°, 360°)
`norm180(x)`	Angle → Normalization	Any angle [°]	Equivalent in (−180°, 180°]
`norm_angle(x)`	Angle → Normalization	Any angle [rad]	Equivalent in [0, 2π)
`complement(x)`	Angle → Relations	Angle [°]	90° − x
`supplement(x)`	Angle → Relations	Angle [°]	180° − x
`anglediff(a, b)`	Angle → Relations	Two angles [°]	Signed shortest difference in (−180°, 180°]

Angle Group

The Angle group collects scalar helpers for unit conversion, range normalization, and basic angle arithmetic. All functions work on plain scalar values and return a single number – they do not produce a named result list.

Conversion – `deg2rad` `rad2deg` `dms2deg` `angpct`

Syntax

deg2rad(x)
rad2deg(x)
dms2deg(d, m, s)
angpct(deg=...)
angpct(pct=...)

Description

Function	Formula	Result
`deg2rad(x)`	x ⋅ π / 180	Angle in radians
`rad2deg(x)`	x ⋅ 180 / π	Angle in degrees
`dms2deg(d, m, s)`	\|d\| + m/60 + s/3600 (sign of d)	Decimal degrees
`angpct(deg=...)`	tan(deg ⋅ π/180) ⋅ 100	Slope grade [%]
`angpct(pct=...)`	atan(pct/100) ⋅ 180/π	Angle in degrees [°]

Examples

deg2rad(180)         = 3.14159…   (= π)
rad2deg(π)           = 180
dms2deg(48, 30, 0)   = 48.5
dms2deg(-7, 30, 0)   = -7.5
angpct(deg=45)       = 100
angpct(pct=12)       = 6.8428…°

Normalization – `norm360` `norm180` `norm_angle`

Syntax

norm360(x)
norm180(x)
norm_angle(x)

Description

Function	Input	Output range	Description
`norm360(x)`	Angle [°]	[0°, 360°)	Maps any degree value into the positive full circle
`norm180(x)`	Angle [°]	(−180°, 180°]	Maps any degree value to the signed half-circle range
`norm_angle(x)`	Angle [rad]	[0, 2π)	Maps any radian value into the positive full circle

Examples

norm360(-90)      = 270
norm360(400)      = 40
norm180(270)      = -90
norm180(-200)     = 160
norm_angle(-π/2)  = 3π/2 ≈ 4.7124

Relations – `complement` `supplement` `anglediff`

Syntax

complement(x)
supplement(x)
anglediff(a, b)

Description

Function	Formula	Result
`complement(x)`	90 − x	Complementary angle [°]
`supplement(x)`	180 − x	Supplementary angle [°]
`anglediff(a, b)`	shortest signed rotation from a to b [°]	(−180°, 180°]

anglediff: always returns the shortest signed arc, so the result lies in (−180°, 180°]. A positive value means a counter-clockwise rotation, a negative value means clockwise.

Examples

complement(30)        = 60
supplement(120)       = 60
anglediff(350, 10)    = 20
anglediff(10, 350)    = -20
anglediff(0, 180)     = 180
anglediff(0, 181)     = -179

← Back to MathBox Guide

RedcrabX – Geometry Functions

Table of Contents

Overview

Point-to-Point Distance – p2dist

Syntax

Description

Formulas

Result List

Examples

Point-to-Line Distance – p2linedist

Syntax

Description

Formulas

Result List

Examples

Line-to-Line Angle – l2angle

Syntax

Description

Formula

Result List

Examples

2D Vector Angle – v2angle

Syntax

Description

Formula

Result List

Examples

3D Vector Angle – v3angle

Syntax

Description

Formula

Result List

Examples

Triangle Angles from Sides – abcangles

Syntax

Description

Formulas

Result List

Examples

Polygon Angle Sum and Single Angles – polyangles

Syntax

Description

Formulas

Result List

Examples

Equilateral Triangle Solver – eqtri

Syntax

Description

Formulas

Result List

Examples

Isosceles Triangle Solver – isotri

Syntax

Description

Formulas

Result List

Examples

Right Triangle Solver – rtri

Syntax

Description

Formulas

Result List

Examples

Pythagoras Right-Triangle Solver – pythag

Syntax

Description

Formulas

Result List

Examples

Isosceles Right-Triangle Solver – isortri

Syntax

Description

Formulas

Result List

Examples

Triangle Area from Base and Height – triarea

Syntax

Description

Formula

Examples

Point-to-Point Distance – `p2dist`

Point-to-Line Distance – `p2linedist`

Line-to-Line Angle – `l2angle`

2D Vector Angle – `v2angle`

3D Vector Angle – `v3angle`

Triangle Angles from Sides – `abcangles`

Polygon Angle Sum and Single Angles – `polyangles`

Equilateral Triangle Solver – `eqtri`

Isosceles Triangle Solver – `isotri`

Right Triangle Solver – `rtri`

Pythagoras Right-Triangle Solver – `pythag`

Isosceles Right-Triangle Solver – `isortri`

Triangle Area from Base and Height – `triarea`

Triangle Area from Three Sides (Heron) – `triheron`

Triangle Area from Base and Two Angles – `triarea2ang`

Square Solver – `square`

Golden Rectangle Solver – `goldrect`

Parallelogram Solver – `para`

Parallelogram Area from Base and Height – `paraaera`

Rectangle Solver – `rect`

Slope / Gradient – `slope`

Point-to-Point Slope – `p2slope`

Circle Solver – `circle`

Circular Sector Solver – `sect`

Circular Segment Solver – `segm`

Sphere Solver – `sphere`

Bicylinder Solver – `bicylinder`