A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping.
The inverted cycloid (a cycloid rotated through 180°) is the curve of fastest descent under gravity.
One arch of a cycloid generated by a circle of radius r has area = 3πr²
The arc length S of one arch of the cycloid = 8r.
credit to Guillermo Bautista http://mathandmultimedia.com/2010/07/30/geogebra-tutorial-constructing-a-cycloid/
