& Tornado Alley
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
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
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:
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