Hypocycloid Calculator

Calculations with hypocycloids. A hypocycloid is the curve that is generated by a point of a small circle, which is rolling inside a large circle. The ratio of the radiuses of the two circles must be an integer. This ratio determines the number of cusps. The astroid is a hypocycloid with four cusps.
Enter at radiuses and number of cusps two values and choose the number of decimal places. Then click Calculate. n must be an integer >2.

	Radius large circle (a):			A hypocycloid with five cusps
	Radius small circle (b):
	Number of cusps (n):
	Chord length (l):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Generation of a hypocycloid.

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

The hypocycloid has convex vertices and concave arcs. If the smaller circle rolls on the outside of the larger circle and not inside it, then an epicycloid is formed, in which the arcs are directed outward and not inward, thus having concave vertices and convex arcs. There is an epicycloid with n=2, but no such hypocycloid.
The symmetry properties of hypocycloid and epicycloid are the same. The hypocycloid is also 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 outward-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 hypocycloid with even n is also point-symmetric about the intersection point of the symmetry axes.

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

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