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SolidAngle[p,{u1,…,ud}]

gives the solid angle at the point p and spanned by the vectors u1,…,ud.

SolidAngle[p,reg]

gives the solid angle subtended by the region reg.

Details
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Basic Examples  
Scope  
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SolidAngle

SolidAngle[p,{u1,…,ud}]

gives the solid angle at the point p and spanned by the vectors u1,…,ud.

SolidAngle[p,reg]

gives the solid angle subtended by the region reg.

Details

  • SolidAngle is also known as planar angle or spherical angle.
  • SolidAngle is typically used to measure the amount of the field of view from a point that an object covers.
  • SolidAngle[p,{u1,…,ud}] is the measure of the intersection of the d-dimensional unit sphere Sphere[p] and the conic hull generated by the vectors u1,…,ud.
  • SolidAngle[p,reg] is the measure of the intersection of the unit sphere centered at p and half‐lines from p through points of the region reg.

Examples

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Basic Examples  (1)

The solid angle at the point {1/2,1/2,0} and spanned by the vectors {0,0,1}, {0,1,1}, {1,1,1} and {1,0,1}:

Wolfram Language code: SolidAngle[{1 / 2, 1 / 2, 0}, {{0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}]
Wolfram Language code: Graphics3D[Arrow[{{1 / 2, 1 / 2, 0}, #}& /@ {{0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}]]

Scope  (2)

Use SolidAngle to find the angle at the point and spanned by the vectors:

Wolfram Language code: SolidAngle[{0, 0, 0}, {{0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}]
Wolfram Language code: Graphics3D[Arrow[{{0, 0, 0}, #}& /@ {{0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}]]

The solid angle subtended by the Cone[{{1,1,1},{0,0,0}}]:

Wolfram Language code: SolidAngle[{0, 0, 0}, Cone[{{1, 1, 1}, {0, 0, 0}}]]
Wolfram Language code: Graphics3D[{Cone[{{1, 1, 1}, {0, 0, 0}}], Red, Point[{0, 0, 0}]}]

Properties & Relations  (5)

SolidAngle[{0,0},{u1,u2}] is the planar angle between the half‐lines from the point p in the direction of u1 and u2:

Wolfram Language code: SolidAngle[{0, 0}, {{1, 0}, {1, 1}}]
Wolfram Language code: PlanarAngle[{{1, 0}, {0, 0}, {1, 1}}]
Wolfram Language code: Graphics[Arrow[{{0, 0}, #}]& /@ {{1, 0}, {1, 1}}]

SolidAngle[{0,0,0},{u1,u2,u3}] is the surface area of the triangle on the unit sphere with corner points :

Wolfram Language code: SolidAngle[{0, 0, 0}, {{1, 0, 1}, {1, 1, 1}, {0, 0, 1}}]

In 2D, SolidAngle[p,Line[{q1,q2}]] is equivalent to PlanarAngle[{q1,p,q2}]:

Wolfram Language code: SolidAngle[{1, 1 / 2}, Line[{{2, 2}, {3, 1}}]] == PlanarAngle[{{2, 2}, {1, 1 / 2}, {3, 1}}]

In 3D, SolidAngle[p,reg] is the surface area of the intersection of the unit sphere centered at p that lies in the infinite cone with vertex p and enclosing reg:

Wolfram Language code: SolidAngle[{0, 0, 0}, Cone[{{1.5, 1.5, 1.5}, {0, 0, 0}}]]
Wolfram Language code: Area[RegionIntersection[Sphere[], Cone[{{1.5, 1.5, 1.5}, {0, 0, 0}}]]]

SolidAngle[p,{u1,…,ud}] is equivalent to PolyhedronAngle[ℛ,p], 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};
Wolfram Language code: {u1, u2, u3} = {{0, 0, 1}, {1, 0, 0}, {0, 1, 0}};
Wolfram Language code: PolyhedronAngle[ℛ, p]
Wolfram Language code: 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), SolidAngle, Wolfram Language function, https://reference.wolfram.com/language/ref/SolidAngle.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_solidangle, author="Wolfram Research", title="{SolidAngle}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/SolidAngle.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_solidangle, organization={Wolfram Research}, title={SolidAngle}, year={2019}, url={https://reference.wolfram.com/language/ref/SolidAngle.html}, note=[Accessed: 13-June-2026]}