Steradian

A steradian is used to measure "solid" angles

A steradian is related to the surface area of a sphere
in the same way a radian is related to the circumference of a circle:

A Radian "cuts out" a length of a circle's circumference equal to the radius.
A Steradian "cuts out" an area of a sphere
equal to (radius)2.

The name steradian is made up from the Greek stereos for "solid" and radian. The SI Unit abbreviation is "sr".

Sphere vs Steradian

  • The surface area of a sphere is 4πr2,
  • The surface area of a steradian is just r2.

So a sphere measures 4π steradians, or about 12.57 steradians. Likewise a steradian is 1/12.57, or about 8% of a sphere.

And because you are measuring an angle, it doesn't matter what size the sphere is, it will always measure 4π steradians.

Example: a sphere with a radius of 1
(called the "unit sphere"):

  • has a surface area of 4π,
  • and a steradian would "cut out" an area of 1.

 

Radiant Intensity

Radiant intensity (how brightly something shines) can be measured in watts per steradian (W/sr).

Example: You measure the light coming from a powerful globe. Your sensor is 50mm × 50mm in size, and if you hold it 2m away it measures 0.1 Watts. What is the radiant intensity in W/sr ?

Answer: At 2m, one steradian would cut through 2×2 = 4 m2 of the sphere.

And because the sensor is relatively small, its flat surface area is approximately the area of sphere that it occupies. So 0.05 × 0.05 = 0.0025m2.

So, one steradian receives about 0.1 W × (4m2/0.0025m2) = 160 W/sr.

In Degrees

Because you can convert from radians to degrees you can also convert from steradians to "square degrees":

A radian is 180/π degrees, or about 57.296°.

A steradian is (180/π)2 square degrees or about 3282.8 square degrees.