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		   radian definition
A Steradian "cuts out" an area of a sphere
equal to (radius)2		   steradian area = r^2

 

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

Sphere vs Steradian

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

sphere area = 4pi, steradian area = 1

Example: The "unit sphere":

  • has a radius of 1
  • has a surface area of 4π,
  • a steradian "cuts 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 (Watts per steradian)?

Answer: At 2m, one steradian cuts 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 we can convert from radians to degrees we 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.

 

Radian Geometry Index