Select three points at random on the circumference of a unit circle and find the distribution of areas of the resulting triangles determined by these three points. The first point can be assigned coordinates (1, 0) without loss of generality. Call the central angles from the first point to the second and third θ_1 and θ_2. The range of θ_1 can be restricted to [0, π] because of symmetry, but θ_2 can range from [0, 2π). Then A(θ_1, θ_2) = 2sin(1/2 θ_1) sin(1/2 θ_2) sin[1/2(θ_1 - θ_2)], so A^_ | = | ( integral_0^π integral_0^(2π) left bracketing bar A right bracketing bar d θ_2 d θ_1)/( integral_0^π integral_0^(2π) d θ_2 d θ_1) | = | 1/(2π^2) integral_0^π integral_0^(2π) left bracketing bar A right bracketing bar d θ_2 d θ_1.

circle line picking | disk triangle picking | line line picking | sphere point picking