# Cumulative Area under a Cycloid versus the Area of Its Rolling Circle

Cumulative Area under a Cycloid versus the Area of Its Rolling Circle

A point on a circle rolling on a straight line without slipping traces out a cycloid. It has long been known that the area under a single arch of the curve is exactly three times the area of the generating circle, but only recently has it been shown that this is a special case of a relationship that holds throughout the cycloid's formation: the ratio of the areas of the cycloidal sector and circular sector remains exactly 3.