Solid Angle Calculator - Calculator Academy

Enter the spherical surface area (the area of a patch on a sphere centered at the observation point) and the radius of that sphere into the calculator to determine the solid angle the patch subtends.

Solid Angle Formula

This calculator uses the standard solid-angle definition for a spherical patch measured on a sphere centered at the observation point. If you know any two of the three quantities, you can solve for the third with the same relationship. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))

\Omega = \frac{A}{r^2}

The same equation can be rearranged when the missing value is the spherical patch area or the radius. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))

A = \Omega r^2

r = \sqrt{\frac{A}{\Omega}}

What the Inputs Mean

Solid angle (Ω): the 3D angular size of the patch as seen from a point, normally expressed in steradians. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))
Spherical surface area (A): the area cut out on the sphere, not just the flat face area of an object. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))
Radius (r): the distance from the observation point to the sphere on which that patch is measured. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))

In SI usage, the steradian is a named dimensionless derived unit. One steradian occurs when the spherical patch area equals the radius squared. ([govinfo.gov](https://www.govinfo.gov/content/pkg/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253/pdf/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253.pdf))

How to Use the Calculator

Enter any two values and leave the unknown field blank. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))
Use matching units so the area corresponds to the radius unit squared, such as square meters with meters or square feet with feet. ([govinfo.gov](https://www.govinfo.gov/content/pkg/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253/pdf/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253.pdf))
Read the result in steradians, or interpret it in square degrees when that unit is more intuitive. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))

Because the calculation is area divided by radius squared, larger patches increase the result and larger radii decrease it. ([govinfo.gov](https://www.govinfo.gov/content/pkg/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253/pdf/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253.pdf))

Quick Reference Values

A full sphere is the ordinary upper reference case for solid angle. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))
```
\Omega_{\text{full sphere}} = 4\pi\,\mathrm{sr} \approx 12.56637\,\mathrm{sr}
```

A hemisphere is exactly half of the full-sphere value.

\Omega_{\text{hemisphere}} = 2\pi\,\mathrm{sr} \approx 6.28319\,\mathrm{sr}

One steradian can be converted to square degrees for sky coverage or viewing-angle comparisons.
```
1\,\mathrm{sr} \approx 3282.80635\,\mathrm{deg}^2
```
The full sphere in square degrees provides a useful reasonableness check for very large results.
```
4\pi\,\mathrm{sr} \approx 41252.96125\,\mathrm{deg}^2
```

Example

If the spherical patch area is 18 square meters and the radius is 3 meters, the solid angle is 2 steradians. In square degrees, that is about 6565.61.

\Omega = \frac{18}{3^2} = 2\,\mathrm{sr}

Input Tips

Use the radius, not the diameter, because the equation depends on radius squared. ([govinfo.gov](https://www.govinfo.gov/content/pkg/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253/pdf/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253.pdf))
Make sure the area is a spherical patch area measured from the observation point. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))
Keep length and area units consistent before interpreting the answer. ([govinfo.gov](https://www.govinfo.gov/content/pkg/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253/pdf/GOVPUB-C13-60b8e508300eac3d65850cd9d112b253.pdf))
For standard geometry problems, the result should run from zero up to the full-sphere value. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))

If a result looks too large, compare it with the full-sphere reference above; if it is negative or greater than a full sphere in an ordinary setup, recheck the area definition, the radius entry, and the units. ([calculator.academy](https://calculator.academy/solid-angle-calculator/))