Path:  physics insights > basics > Solid Angles

The ordinary angle, measured in radians, is the length of the arc of a unit circle subtended by the angle.  A full circle, 360 degrees, is 2 π radians, of course -- the length of the perimeter of a unit circle.

Solid angles are measured in "steradians"; instead of the arc length of the portion of the unit circle subtended by the angle, it's the area of the unit sphere subtended by the solid angle.

The solid angle subtended by a cone whose apex has angle 2θ is obtained by making the cone 1 unit high, setting it inside the unit sphere, and finding the area of the part of the sphere the cone is sitting on.  More simply, to find the area of a part of a sphere of radius R subtended by a particular (linear) angle, we just integrate (see figure 1 below):

(1)    

The solid angle subtended by the cone with apex 2θ is the area of the unit sphere subtended by the cone, so we just set R=1, to obtain

(2)     

For theta = π, which would include the entire sphere, (2) evaluates to 4π -- and so we see there are 4π steradians in a full sphere.

For a small angle, we can use the second order approximation to cos(θ)

(3)    

and then the solid angle subtended by the cone is approximately

(4)    

This comes up in optics, where we want to know what fraction of the light of a point source is received by a pupil which has angular radius θ when viewed from the location of the point source.  In general, it's the solid angle subtended by the pupil, divided by the solid angle of the entire sphere.  That's

(5)    

or, for small angles, it's approximately

(6)  

Figure 1: Integrating over part of a sphere

Page created on 9/22/2007