# 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 sphereequal to (radius) ^{2}. |

**sr**

*stereos*for "solid" and radian.

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

**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 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 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.