Length of 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 length of one arc of the cycloid (i.e. where $0 \le \theta \le 2 \pi$) is $8a$.

Proof
Let $L$ be the length of one arc of the cycloid.

From Arc Length for Parametric Equations:


 * $\displaystyle L = \int_0^{2\pi} \sqrt {\left({\frac{\mathrm d x}{\mathrm d \theta}}\right)^2 + \left({\frac{\mathrm d y}{\mathrm d \theta}}\right)^2} \mathrm d \theta$

where, from the definition of the cycloid:
 * $x = a \left({\theta - \sin \theta}\right)$
 * $y = a \left({1 - \cos \theta}\right)$

we have:

Thus:

Thus:

So $L = 8a$ where $a$ is the radius of the generating circle.

Also see

 * Area under Arc of Cycloid