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Steradian — Definition, Formula & Examples

A steradian (sr) is the SI unit of solid angle — the three-dimensional analog of the radian. One steradian is the solid angle that, from the center of a sphere, subtends an area on the surface equal to the square of the sphere's radius.

A steradian is defined as the solid angle Ω\Omega subtended at the center of a sphere of radius rr by a surface patch of area A=r2A = r^2. Since the total surface area of a sphere is 4πr24\pi r^2, a complete sphere subtends exactly 4π4\pi steradians.

Key Formula

Ω=Ar2\Omega = \frac{A}{r^2}
Where:
  • Ω\Omega = Solid angle in steradians (sr)
  • AA = Area on the sphere's surface subtended by the solid angle
  • rr = Radius of the sphere

How It Works

Solid angle measures how large an object appears from a given point, extending the idea of a planar angle into three dimensions. You calculate it by dividing the area of the surface patch (projected onto a sphere centered at the point) by the square of the radius. A hemisphere subtends 2π2\pi sr, and a full sphere subtends 4π≈12.5664\pi \approx 12.566 sr. Steradians are dimensionless in the same way radians are — they are ratios of like-dimensioned quantities — but carry the unit symbol sr for clarity.

Worked Example

Problem: A spotlight illuminates a circular patch of area 9 m² on the inside of a dome with radius 3 m. What solid angle does the patch subtend at the center of the dome?
Identify values: The surface area is A=9A = 9 m² and the radius is r=3r = 3 m.
Apply the formula: Divide the area by the square of the radius.
Ω=Ar2=932=99=1 sr\Omega = \frac{A}{r^2} = \frac{9}{3^2} = \frac{9}{9} = 1 \text{ sr}
Answer: The spotlight subtends a solid angle of 1 steradian.

Why It Matters

Steradians appear throughout physics and engineering whenever intensity is measured per unit solid angle. Luminous intensity (candela), radiant intensity, and antenna gain all depend on solid angle. Understanding steradians is essential in optics, astrophysics, and radiometry courses.

Common Mistakes

Mistake: Assuming a full sphere subtends 2π2\pi steradians (by analogy with a full circle being 2π2\pi radians).
Correction: A full sphere subtends 4π4\pi steradians because the total surface area of a sphere is 4πr24\pi r^2, giving Ω=4πr2/r2=4π\Omega = 4\pi r^2 / r^2 = 4\pi sr.