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}. |
Sphere vs Steradian
- The surface area of a sphere is 4πr^{2},
- The surface area of a steradian is just r^{2.}
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 m^{2} 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 = 0.0025m^{2}.
So, one steradian receives about 0.1 W × (4m^{2}/0.0025m^{2}) = 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.