The key insight: Drag the orange radius handle outward on either panel — watch the arc length / cap area scale up, but the computed angle stays exactly the same. That's because θ = s/r and Ω = A/r² are pure dimensionless ratios — the "unit" (radian or steradian) is really just a reminder of which ratio you computed, not a physical dimension. Try dragging the arc endpoint (cyan) around the full circle to see θ go from 0 to 2π.
Plane Angle — θ = s / r
Drag the cyan arc handle · drag the orange radius handle
DRAG HANDLES
Arc length s
—r-units
s = r · θ = —
Radius r
—units
drag orange handle
Plane angle θ = s / r
—rad
= — · π | —°
Solid Angle — Ω = A / r²
Drag the violet cap handle · drag the orange radius handle
DRAG HANDLES
Cap area A
—r²
A = 2πr²(1−cosα) = —
Radius r
—units
drag orange handle
Solid angle Ω = A / r²
—sr
= 2π(1−cosα) | half-angle α = —°
Plane Angle — Radian
θ (rad) = s / r
s = arc length · r = radius — both in the same length unit, so the ratio is dimensionless.Full circle: s = 2πr → θ = 2π ≈ 6.283 rad
1 radian is the angle subtended when s = r exactly.
To convert: degrees × π/180 = radians
Solid Angle — Steradian
Ω (sr) = A / r²
A = spherical cap area = 2πr²(1−cosα) · α = half-angle of the cone.Full sphere: A = 4πr² → Ω = 4π ≈ 12.566 sr
1 steradian subtends area r² on the sphere surface.
Hemisphere: α = 90° → Ω = 2π ≈ 6.283 sr