Solid Angle - Definition and Properties

Definition and Properties

An object's solid angle is equal to the area of the segment of a unit sphere, centered at the angle's vertex, that the object covers. A solid angle equals the area of a segment of unit sphere in the same way a planar angle equals the length of an arc of a unit circle.

The solid angle of a sphere measured from a point in its interior is 4π sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2π/3 sr. Solid angles can also be measured in square degrees (1 sr = (180/π)2 square degree) or in fractions of the sphere (i.e., fractional area), 1 sr = 1/4π fractional area.

In spherical coordinates, there is a simple formula as

The solid angle for an arbitrary oriented surface S subtended at a point P is equal to the solid angle of the projection of the surface S to the unit sphere with center P, which can be calculated as the surface integral:

where is the vector position of an infinitesimal area of surface with respect to point P and where represents the unit vector normal to . Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product .