Area under Arc of Cycloid

Theorem
Let $C$ be a cycloid generated by the equations:
 * $x = a \paren {\theta - \sin \theta}$
 * $y = a \paren {1 - \cos \theta}$

Then the area under one arc of the cycloid is $3 \pi a^2$.

That is, the area under one arc of the cycloid is three times the area of the generating circle.

Proof
Let $A$ be the area under of one arc of the cycloid.

From Area under Curve, $A$ is defined by:

But:
 * $\dfrac {\d x} {\d \theta} = a \paren {1 - \cos \theta}$

and so:

Also see

 * Length of Arc of Cycloid