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PolyhedronAngle[poly,p]

gives the solid angle at the point p and spanned by edges with common point p.

PolyhedronAngle[poly,e]

gives the dihedral angle between the two faces with common edge e.

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PolyhedronAngle

PolyhedronAngle[poly,p]

gives the solid angle at the point p and spanned by edges with common point p.

PolyhedronAngle[poly,e]

gives the dihedral angle between the two faces with common edge e.

Details

Examples

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Basic Examples  (2)

The dihedral angle between two faces with common edge {1,2} of ℛ in canonical form:

Wolfram Language code: ℛ = Polyhedron[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}, {{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}}];
Wolfram Language code: PolyhedronAngle[ℛ, {1, 2}]

The solid angle at the vertex point {0,0,0} and spanned by edges:

Wolfram Language code: ℛ = Polyhedron[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}, {{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}}];
Wolfram Language code: PolyhedronAngle[ℛ, {0, 0, 0}]

The solid angle of ℛ at the point 1 in canonical form:

Wolfram Language code: PolyhedronAngle[ℛ, 1]

Scope  (2)

Use PolyhedronAngle to find the solid angles at the vertex points:

Wolfram Language code: ℛ = Polyhedron[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}, {{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}}];
Wolfram Language code: PolyhedronAngle[ℛ, #]& /@ PolyhedronCoordinates[ℛ]

The dihedral angle at the given edge:

Wolfram Language code: PolyhedronAngle[ℛ, {1, 2}]

PolyhedronAngle works on polyhedra:

Wolfram Language code: Polyhedron[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{2, 3, 4}, {3, 2, 1}, {4, 1, 2}, {1, 4, 3}}];
Wolfram Language code: PolyhedronAngle[%, 1]

Tetrahedron:

Wolfram Language code: PolyhedronAngle[Tetrahedron[], 1]

Octahedron:

Wolfram Language code: PolyhedronAngle[Octahedron[], 1]

Properties & Relations  (2)

PolyhedronAngle[ℛ,{p1,p2}] is equivalent to DihedralAngle[{p1,p2},{v,w}] where v and w are vectors in adjacent faces of {p1,p2} in a polyhedron ℛ:

Wolfram Language code: ℛ = Polyhedron[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}, {{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}}];
Wolfram Language code: p1 = {0, 0, 0};p2 = {0, 1, 0}; {v, w} = {{0, 0, 1}, {1, 0, 0}} - {p1, p1};
Wolfram Language code: {PolyhedronAngle[ℛ, {p1, p2}], DihedralAngle[{p1, p2}, {v, w}]}
Wolfram Language code: Graphics3D[{{Opacity[.5], Green, ℛ}, Thickness[.05], Red, Line[{{p1, p1 + v}, {p1, p1 + w}}], Blue, Line[{p1, p2}]}]

PolyhedronAngle[ℛ,p] is equivalent to SolidAngle[p,{u1,…,ud}] where u1,…,ud are vectors adjacent to the point p in a polyhedron ℛ:

Wolfram Language code: ℛ = Polyhedron[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}, {{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}}];
Wolfram Language code: p = {0, 0, 0};{u1, u2, u3} = {{0, 0, 1}, {1, 0, 0}, {0, 1, 0}};
Wolfram Language code: {PolyhedronAngle[ℛ, p], SolidAngle[p, {u1, u2, u3}]}
Wolfram Language code: Graphics3D[{{Opacity[.5], Green, ℛ}, Thickness[.05], Red, Line[{p, #}& /@ {u1, u2, u3}], Blue, Sphere[p, .1]}]
Wolfram Research (2019), PolyhedronAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/PolyhedronAngle.html.

Text

Wolfram Research (2019), PolyhedronAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/PolyhedronAngle.html.

CMS

Wolfram Language. 2019. "PolyhedronAngle." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PolyhedronAngle.html.

APA

Wolfram Language. (2019). PolyhedronAngle. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolyhedronAngle.html

BibTeX

@misc{reference.wolfram_2026_polyhedronangle, author="Wolfram Research", title="{PolyhedronAngle}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/PolyhedronAngle.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_polyhedronangle, organization={Wolfram Research}, title={PolyhedronAngle}, year={2019}, url={https://reference.wolfram.com/language/ref/PolyhedronAngle.html}, note=[Accessed: 13-June-2026]}