Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as (1, 0), and the second by (cos θ, sin θ), with θ element [0, π] (by symmetry, the range can be limited to π instead of 2π). The distance s between the two points is then s(θ) = sqrt(2 - 2cos θ) = 2 left bracketing bar sin(1/2 θ) right bracketing bar . The average distance is then given by s^_ = ( integral_0^π s(θ) d θ)/( integral_0^π d θ) = 4/π.
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