# Length of Arc of Cycloid

## Theorem

Let $C$ be a cycloid generated by the equations:

- $x = a \paren {\theta - \sin \theta}$
- $y = a \paren {1 - \cos \theta}$

Then the length of one arc of the cycloid is $8 a$.

## Proof 1

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:

\(\displaystyle \frac {\d x} {\d \theta}\) | \(=\) | \(\displaystyle a \paren {1 - \cos \theta}\) | |||||||||||

\(\displaystyle \frac {\d y} {\d \theta}\) | \(=\) | \(\displaystyle a \sin \theta\) |

Thus:

\(\displaystyle \paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2\) | \(=\) | \(\displaystyle a^2 \paren {\paren {1 - \cos \theta}^2 + \sin^2 \theta}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a^2 \paren {1 - 2 \cos \theta + \cos^2 \theta + \sin^2 \theta}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 a^2 \paren {1 - \cos \theta}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 4 a^2 \map {\sin^2} {\theta / 2}\) | Half Angle Formula for Sine |

Thus:

\(\displaystyle L\) | \(=\) | \(\displaystyle \int_0^{2 \pi} 2 a \, \map \sin {\theta / 2} \rd \theta\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \bigintlimits {-4 a \, \map \cos {\theta / 2} } 0 {2 \pi}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 8 a\) |

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

$\blacksquare$

## Proof 2

Consider the tangent $PQ$ to both the generating circle and the cycloid itself.

By

:

- $PR = 2 PQ$

In the limit, where $P$ is at the cusp, the tangent $PQ$ is perpendicular to the straight line along which the generating circle rolls.

At this point:

- $PQ = 2 a$.

Thus at this point:

- $PR = 4 a$

But $4 a$ is half the length of one arc of $C$.

Hence the result.

## Also see

## Historical Note

The geometric proof of the length of the arc of a cycloid was demonstrated by Christopher Wren in $1658$.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Cycloid: $11.6$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.6$: The Brachistochrone. Fermat and the Bernoullis