Solid Angle
The three-dimensional version of an angle.
Lesson at a glanceShow the key points
- θ = l/r; Ω = S/r² (steradian), independent of r.
- Small tilted patch: ΔΩ = ΔS cos α / r².
- Closed curve: 2π at an inside point. Closed surface: 4π inside, 0 outside.
- A cone of half-angle α: Ω = 2π(1 − cos α).
A plane angle measures how wide a pair of lines opens at a point. A solid angle does the same in three dimensions: how wide a cone opens. It is the tool that turns Coulomb’s law into Gauss’s law.
From plane angles to solid angles
Join the two ends of a curve to a point O and the curve subtends a plane angle at O. To measure it, draw a circle of any radius r about O and measure the arc length l cut off: θ = l/r.
Now take a surface patch and join every point on its edge to O. The lines form a cone (like a paper cone for roasted peanuts), and we say the patch subtends a solid angle at O. Measure it the same way, with a sphere instead of a circle:
A small tilted patch
A short segment Δl at distance r, square-on to the line from O, subtends Δθ ≈ Δl/r. If the line from O makes an angle α with the segment’s normal, only the part Δl cos α faces O, so Δθ = Δl cos α / r. The same holds for a small area:
A small tile of area 4.0 cm² is 2.0 m from a point O. The line from O to the tile makes 60° with the tile’s normal. What solid angle does it subtend at O?
Whole circles and whole spheres
A full circle subtends θ = 2πr/r = 2π at its centre, and in fact any closed curve subtends 2π at any point inside it. A full sphere subtends Ω = 4πr²/r² = 4π, and any closed surface subtends 4π at any point inside.
What about a point outside? Close a curve around an outside point step by step and the angle it subtends shrinks to zero: the near and far parts of the curve cover the same directions with opposite sense and cancel. So a closed surface subtends zero solid angle at an external point.
What solid angle does a cone of half-angle 90° (a flat disc, seen from its centre) subtend? And one of half-angle 60°?
- 1Plane angle θ = l/r; solid angle Ω = S/r², dimensionless (steradian).
- 2Small tilted patch: ΔΩ = ΔS cos α / r².
- 3A closed surface subtends 4π at an inside point and 0 at an outside point.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.