Epicycloid Calculator

Calculations with epicycloids. These figures are the path of a small circle rolling around a larger one. So, an epicycloid is generated like a hypocycloid, but with the smaller (or equal, at n=1) circle rolling around the fixed circle on the outside. An epicycloid with one arc is a cardioid, with two arcs it is a nephroid.
Enter at radiuses and number of arcs two values and choose the number of decimal places. Then click Calculate. n must be a natural number.

	Radius large circle (a):			A nephroid, n=2 An epicycloid with n=3 Generating an epicycloid
	Radius small circle (b):
	Number of arcs (n):
	Diameter (d):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.
	Formulas: $$n=a/b$$ $$d=2\cdot a+4\cdot b$$ $$p=8\cdot (n+1)\cdot b$$ $$A=(n+1)\cdot (n+2)\cdot b^2\cdot \pi$$ pi: $$\pi =\mathrm{3.141592653589793...}$$

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

The epicycloid is axially symmetric about a number of axes equal to the number of arcs n. For an even number n, there are two different types of axes of symmetry: those passing through the center of two opposite arcs and those passing through two opposite inward-pointing vertices. For an odd number n, there is only one type of axis of symmetry, which passes through the center of an arc and through the directly opposite vertex. It is also rotationally symmetric about this point at an angle of 360°/n and multiples thereof. The epicycloid with even n is also point-symmetric about the intersection point of the symmetry axes.
The Ptolemaic worldview assumed that the planets orbited the Earth in epicyclic orbits. This was necessary to explain the observed positions of celestial bodies. The realization that the planets revolved around the Sun, not the Earth, only slowly gained acceptance from the 16th century onward. Epicycloids were studied mathematically in the early 18th century by the French mathematician Philippe de La Hire.

Last updated on 03/30/2026. Author: Jürgen Kummer

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