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Solid Angle — Definition, Formula & Examples

A solid angle is the three-dimensional analog of a regular (planar) angle — it measures how large an object appears from a given point by describing the size of the cone of directions from that point to the object. Solid angles are measured in steradians (sr).

The solid angle Ω\Omega subtended by a surface SS at a point PP is defined as the area of the projection of SS onto a unit sphere centered at PP. Equivalently, Ω=∫Sr^⋅dAr2\Omega = \int_S \frac{\hat{r} \cdot d\mathbf{A}}{r^2}, where rr is the distance from PP to each surface element and r^\hat{r} is the unit radial vector. The SI unit is the steradian (sr), and a full sphere subtends 4π4\pi sr.

Key Formula

Ω=2π(1−cos⁡θ)\Omega = 2\pi\left(1 - \cos\theta\right)
Where:
  • Ω\Omega = Solid angle in steradians (sr)
  • θ\theta = Half-angle of the cone measured from its axis

How It Works

Just as a planar angle in radians equals the arc length cut on a unit circle, a solid angle in steradians equals the area cut on a unit sphere. To find the solid angle subtended by a surface, you project that surface onto a sphere of radius rr centered at your viewpoint, compute the projected area AA, and divide by r2r^2. A hemisphere subtends 2π2\pi sr, and the entire sphere subtends 4π≈12.5664\pi \approx 12.566 sr. For a right circular cone with half-angle θ\theta, the solid angle has a clean closed-form expression.

Worked Example

Problem: Find the solid angle subtended by a cone with a half-angle of 60°.
Step 1: Identify the half-angle and convert if needed.
θ=60°\theta = 60°
Step 2: Apply the cone solid-angle formula.
Ω=2π(1−cos⁡60°)=2π(1−12)=2π⋅12=π sr\Omega = 2\pi(1 - \cos 60°) = 2\pi\left(1 - \frac{1}{2}\right) = 2\pi \cdot \frac{1}{2} = \pi \text{ sr}
Step 3: Interpret the result: this cone covers π/(4π)=25%\pi / (4\pi) = 25\% of the full sphere.
Ω4π=π4π=0.25\frac{\Omega}{4\pi} = \frac{\pi}{4\pi} = 0.25
Answer: The cone subtends a solid angle of π≈3.14\pi \approx 3.14 steradians, which is 25% of the full sphere.

Why It Matters

Solid angles appear in physics whenever you need to quantify how much of a surrounding space a source or detector covers. In radiometry and photometry, luminous intensity is defined as luminous flux per steradian. In astrophysics, the apparent size of celestial objects is given in square degrees or steradians.

Common Mistakes

Mistake: Confusing the solid angle of a full sphere (4π4\pi sr) with 2π2\pi sr.
Correction: A hemisphere subtends 2π2\pi sr. The complete sphere subtends 4π4\pi sr, since the total surface area of a unit sphere is 4π4\pi.