Expected chord between two random points

HardProbability Expected Value~7m

Two points are chosen independently and uniformly on the circumference of a unit circle. Find the expected length of the chord joining them. The answer has the form $\dfrac{x}{\pi}$ for a rational $x$ ; find $x$ .

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By symmetry, only the angular separation $\Theta \sim \text{Uniform}(0,2\pi)$ matters.
A chord subtending angle $\Theta$ on a unit circle has length $2\sin(\Theta/2)$ . Integrate against the uniform density.

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Expected chord between two random points

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