Length of Arc of Cycloid/Proof 1

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

From Arc Length for Parametric Equations:


 * $\displaystyle L = \int_0^{2 \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$

where, from Equation of Cycloid:
 * $x = a \paren {\theta - \sin \theta}$
 * $y = a \paren {1 - \cos \theta}$

we have:

Thus:

Thus:

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