First, let's review solid angle.

It's a concept that not everyone uses.

And we are going to need to be able to calculate it easily.

So, here's a sphere, we have a certain radius of that sphere.

There is a cone,

and the cone has a certain patch or area on the surface of the sphere,

and has a central half angle of theta.

So, the Radian is defined as theta,

as the arc that would describe the radius of this little circle here,

a brown area A,

over the radius of the sphere.

And of course, you know that the circumference of an arc related to the radius,

there would be a total of two pi possible radians.

The total distance around the circle would be two pi times r,

divided by r. Solid angle is the obvious analogy to two dimensions.

So, we take the area of our spherical cap and we divide by r squared.

And if you are in the area of a sphere,

or pi r squared, that tells you that a total sphere,

the biggest solid angle you can have,

and we do have light sources that do this,

would radiate into a solid angle of four pi steradians.

By the way, the word ster there it comes from stereo,

which from the Greek means solid.

So that is why it is solid angle,

it is just the translation, so it is a little bit of calculation.

So, here we have the tools we need.

By definition, solid angle is the area over the radius of the sphere squared.

Well, we can do just our standard trig here to know what the area of that little cone is,

integrated in two angular coordinates from the spherical coordinate system.

We can do that integral,

and we find overall that the solid angles of the patch is four pi,

makes sense, there is our maximum value,

times the sine squared of the half angle over two.

It's just the second trig identity here.

So, that's a convenient equation to have

that tells you how to get between angle and radians,

and solid angle and steradians,

for arbitrary angle theta.

It's convenient to expand this in its lowest order for small angles.

And so, for small angles,

where we can take the first order approximation of sine,

there is simply a quadratic relationship between the angle and radians,

and the solid angle and steradians.

So we'll need that math because we're going to be calculating steradians

in a fair amount as we go through the rest of this module.