How do you measure a solid angle?

Answer 1

A 'solid angle' is a convenience term. I actually describes the area on the surface of a sphere rather than the distance of an arc segment (as does a planar angle)

Since the surface area of a sphere of radius r is given by

A = 4*π*r²

we say that a sphere subtends a solid angle of 4*π 'steradians'. (This is very much like saying that a full circle subtends a planar angle of 2π radians)

If a unit sphere (having a radius of 1) has an area on its surface that is equal to 1/8 of the total surface area, then we say that area subtends a solid angle of π/2 'steradians'

But this says nothing about the 'shape' of the surface area. It can be any odd shape you like.

The way to 'measure' a solid angle is to measure the area it subtends on teh surface of a sphere, divide that by the total area of the sphere, and multiply that by 4π.

Hope that helps.

Doug

Answer 2

Explanation motivated from Optics.

Suppose you have a laser with a 10 degree half angle divergence. The fraction of the surface area of a sphere traced out is:

A = 2πr^2( 1 - cos(θ)) where θ is the half angle divergence

By definition the steradian is: A/r^2 ≡ Ω

So in my spherical example:

Ω = 2π( 1 - cos(θ))

As someone pointed out above, a sphere subtends 4π steradians, a point source that radiates in all directions subtends 4π steradians, a disk subtends 2π steradians and my 10 degree laser example subtends about 0.095 steradians.

Also, as pointed out above, the shape of the area does not matter. I simply used a spherical example to help explain the explanation.

Answer 3

With a protractor? You're gonna have to be more specific.