# Solid angle

A part of space bounded by all the half-lines going from one point (the vertex) through the points of a closed curve (see Fig. a, Fig. b).

Figure: s086060a

Figure: s086060b

A particular case of a solid angle is that of a polyhedral angle. One takes as measure of a solid angle the ratio of the surface cut out by it on a sphere centred at the vertex to the squared radius of this sphere. For example, the solid angle containing $1/8$ of the space (an octant) is measured by the number $4\pi R^2/8R^2=\pi/2$. The unit of measurement of solid angles is the steradian.

#### Comments

The solid angle represents the visual angle under which the curve is seen from the vertex.

#### References

"Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps", http://arxiv.org/abs/1205.1396

**How to Cite This Entry:**

Solid angle.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Solid_angle&oldid=31702