# Solid Angle

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## solid angle

[′säl·əd ′aŋ·gəl]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Solid Angle

a surface formed by rays having a common origin and passing through a closed curve (Figure 1). Sometimes, particularly in Soviet usage, the term “solid angle” is applied to the portion of space bounded by such a surface. Trihedral and polyhedral angles are special cases of solid angles.

A measure of the solid angle subtended at a given point by a surface *S* is provided by the ratio *AIR*^{2}. Here, *A* is the area of the portion of a sphere, with center at the given point, that is cut by a conical surface with vertex at the point and with the perimeter of *S* as a directrix, and *R* is the radius of the sphere. Clearly, the measure of a solid angle is an abstract number. For example, the measure of a solid angle that encloses an octant, that is, oneeighth of space, is the number 4π *R*^{2/}8 *R*^{2} = π/2 (*Figure 2*).

The unit of measure of a solid angle is called the steradian. It is equal to the solid angle subtended at the center of a sphere of unit radius by a portion of the sphere’s surface that is of unit area. The total solid angle about a point is equal to 4π steradians.