Cardioid Calculator

Calculations at a cardioid (heart-shaped curve), an epicycloid with one arc. A cardioid is formed by a circle of the diameter a, which adjacently rolls around another circle of the same size. The trace of one point on the rolling circle produces this shape. The cardioid has the diameter 2a on its symmetry axis.
Enter one value and choose the number of decimal places. Then click Calculate.

	Radius of the circle (r):
	Diameter of the circle (a):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Radius, diameter and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The length of a cardioid, here its perimeter, was first calculated in 1708 by the French mathematician Philippe de La Hire. The seemingly mathematically simple relationship that this length corresponds to eight times the diameter or 16 times the radius of each of the two underlying circles, is actually not quite so simple in its derivation and requires integration. The term cardioid for this shape was established some years after Philippe de La Hire's calculation by the Italian mathematician Giovanni Francesco Salvemini de Castillon. In fact, this is probably slightly more reminiscent of the shape of a human heart than the usual stylized heart shape with a convex tip.
A natural cardioid as a light phenomenon can be seen on the bottom of a cylindrical cup, where light falls laterally onto one of the inner surfaces and is reflected onto the bottom of the cup.
The cardioid is axially symmetrical to the line through its vertex and the opposite point on the arc farthest from its vertex. It has no other symmetries.