Length of Arc of Cycloid/Proof 2

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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 $8 a$.


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




$PR = 2 PQ$

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.

Historical Note

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