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 hdr

 

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 full sphere has a surface area of 4π steradians which is about 12.57 steradians. That means one steradian covers roughly 8% of the sphere.

And because we are measuring angles, 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.

You hold a 50 mm × 50 mm sensor 2 m away and it measures 0.1 Watts.

What is the radiant intensity in W/sr (Watts per steradian)?

  • At 2 meters, one steradian covers an area of (2 m)2 = 4 m2
  • The sensor is small, so the area of sphere it occupies is about equal to its flat surface area: (0.05 m)2 = 0.0025 m2
  • Scale up the measured power: 0.1 W × 4 m20.0025 m2 = 160 W/sr

Square Degrees

Because we can convert from radians to degrees we can also convert from steradians to square degrees (deg2):

Since a radian is (180/π) degrees, we can square this to find that a steradian is (180/π)2 ≈ 3282.8 square degrees

full moon

Example: The Moon

The Moon's angular diameter is about 0.5°, so it covers:

π4 x (0.5°)2
≈ 0.2 deg2

That is a tiny fraction of the visible sky, which covers:

360° × 180°
≈ 64,800 deg2

 

Radian Geometry Index