Cycloid - Arc Length

Arc Length

The arc length S of one arch is given by

\begin{align} S &= \int_{t=0}^{t=2 \pi} \left^\frac{1}{2} dt \\ &= \int_{t=0}^{t=2 \pi} r \sqrt{2 - 2\cos(t)} dt \\ &= \int_{t=0}^{t=2 \pi} 2r \sin\left(\frac{t}{2}\right) dt \\ &= 8 r \end{align}

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