Thayer Watkins
Silicon Valley
& Tornado Alley

The Concept of Solid Angle

To people aquainted with ordinary angles the concept of solid angle and the term steradian are a bit mysterious if not perplexing. We know what a 90° angle looks like and even if 90° is called π/2 radians we are not troubled. With solid angles it is a different matter.

The problem lies in the fact that an ordinary angle can be conceived of without reference to an arbitrary reference circle but a solid angle cannot be properly understood without reference to an arbitrary sphere.

First let us consider an ordinary angle and a reference circle. The angle θ may be defined in terms of the arc shown below. Let s be the length of that arc and r the radius of the circle.

The angle in radians is given by

θ = (s/r)
or, in degrees
θ = (360/2π)(s/r)

Now consider a cone which intersects the sphere of radius R. Let S be the area of surface subtended by the intersection of the cone and the sphere.

The solid angle is defined to be

Ω = (S/r²)

This is the solid angle in steradians. If the surface covers the whole sphere then the number of steradians is 4π.

If one knows the solid angle Ω in steradians then the area of the surface of intersection for any sphere of radius R is given by:

S = R²Ω

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