Solid Angle Calculator (Steradian)
Convert between steradians, square degrees, and apex angles.
Compute solid angle subtended by a cone or by an area at a distance for optics and astronomy.
Solid Angle
A solid angle measures how big a 3-D “cone of view” is, from a single vantage point. It is the 3-D analogue of a planar angle in radians, and its SI unit is the steradian (sr).
Total Sphere
The full sphere subtends 4π steradians ≈ 12.566 sr. A hemisphere is exactly 2π sr.
Cone of Half-Angle θ
For a right circular cone with full apex angle 2θ:
Ω = 2π × (1 − cos(θ)) sr
where θ is the half-angle (axis to edge). For very narrow cones, this simplifies to Ω ≈ π × θ² (with θ in radians).
Area at Distance
For a small area A on a sphere of radius r, or any small area perpendicular to the line of sight:
Ω = A / r²
This is the working definition: a unit steradian is the solid angle subtended by an area equal to r² at distance r.
Square Degrees
Astronomers often use the unit square degree:
1 sr = (180/π)² ≈ 3282.806 deg² Whole sphere = 41,253 deg²
Common Reference Angles
| Source | Angular size | Solid angle |
|---|---|---|
| Sun from Earth | 0.266° half-angle | 6.8 × 10⁻⁵ sr |
| Moon from Earth | 0.259° half-angle | 6.4 × 10⁻⁵ sr |
| Hubble Deep Field | 5.3 arcmin² ≈ 1.5 × 10⁻³ deg² | 4.5 × 10⁻⁷ sr |
| Andromeda galaxy | 1.5° × 0.5° | ~ 0.75 deg² |
| Milky Way disc (±5° band) | wide | ~ 3600 deg² |
The Hubble Deep Field row is the one worth pausing on. That entire famous image, with its several thousand galaxies, covers about 1/24,000,000 of the sky. Hold a grain of sand at arm’s length and you have blocked out roughly that much of it.
Applications
| Field | Use |
|---|---|
| Optics | Lens collection efficiency, étendue |
| Astronomy | Sky-survey areas, intensity from sources |
| Radiometry | Radiant intensity (W/sr) and radiance (W/m²/sr) |
| Acoustics | Directionality of speakers / mics |
| Computer graphics | Hemispherical lighting integrals |
Worked Example: A 10° Beam Flashlight
Half-angle θ = 5° = 0.0873 rad.
- Ω = 2π × (1 − cos(5°)) = 2π × (1 − 0.9962) = 0.0239 sr
- That is about 78.4 deg²: roughly the size of a postage stamp held at arm’s length.
Worked Example: A Window 1 m² Seen from 4 m
Ω = A / r² = 1 / 16 = 0.0625 sr (≈ 205 deg²). This is valid as long as the area is small relative to the distance. For wide views you need the cone formula instead.
Caveats
The Ω = A/r² formula is exact only when the area sits perpendicular to the line of sight and is small enough that the sphere can be approximated as flat in that region. For oblique surfaces, multiply by cos(angle from normal). That factor is the basis of Lambert’s cosine law in optics and radiometry.
A quick feel for when the shortcut stops working, taking a flat disc as the test case. At A/r² = 0.01 sr the flat answer overstates the true solid angle by 0.24%. At 0.03 sr it is 0.7% high, at 0.1 sr it is 2.4% high, and at 0.5 sr it is 11.8% high. Past about 0.03 sr you are trading real accuracy for arithmetic convenience.
Worth being honest about the window example above: 0.0625 sr overstates the exact figure by 1.5%. That is fine for working out roughly how much daylight a room gets, and not fine if you are calibrating an instrument. The calculator reports the equivalent cone half-angle alongside the area answer so you can compare the two directly.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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