Area of Epicycloid and Hypocycloid

curve:

epicycloid

hypocycloid

radius a

1

slices

10

roll

0

show radii

show curve

This Demonstration shows that the area under the first hump of a epicycloid is

(3+2/a)π

when the radii of the generating circle and greater circle are

1

and

a

respectively. When you slide the "roll" slider, slices form a circle of radius

2+1/a

and a circular hole of radius

1+1/a

. Therefore the area is the difference of areas of the two circles. In other words,

2

(2+1/a)

π-

2

(1+1/a)

π=(3+2/a)π

.

For the hypocycloid, the same result holds with

a

negative.

As the number of slices goes to infinity, the dark figure approaches a region bounded by a perfect epicycloid or hypocycloid.