Cycloid - Arc Length

Arc Length

The arc length S of one arch is given by

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

Famous quotes containing the words arc and/or length:

“If I am not, may God put me there; and if I am, may God so keep me.”
—Joan Of Arc (c.1412–1431)

“All expression of truth does at length take this deep ethical form.”
—Henry David Thoreau (1817–1862)