Area under Arc of Cycloid

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

Then the area under one arc of the cycloid (i.e. where $0 \le \theta \le 2 \pi$) 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. Then:

The area required is defined by:

But:
 * $\dfrac{\mathrm d x}{\mathrm d \theta} = a \left({1 - \cos \theta}\right)$

and so:

Also see

 * Length of Arc of Cycloid