Epicycloid
The flower of cusps traced by a circle rolling outside a fixed circle; k = R/r sets the cusp count.
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Formulas and key values
- Parametric equations
- Cusps · whole-number kIn the graph = 5
- Perimeter · whole-number kIn the graph = 9.6
- AreaIn the graph ≈ 5.278
R is the fixed circle’s radius (R = 1 in the graph), r the rolling circle’s, and k = R/r.
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Start the video01 · Definition
What is the Epicycloid?
Roll a circle of radius r around the outside of a fixed circle of radius R, and a rim point traces the epicycloid. When the ratio k = R/r is an integer the curve closes after one lap with exactly k outward-pointing cusps: k = 1 gives the cardioid, k = 2 the nephroid.
Epicycles — circles rolling on circles — carried planetary astronomy from Ptolemy to Copernicus, and the same geometry now shapes the flanks of cycloidal gears in clocks and reducers. The cusps are where the tracing point momentarily touches the base circle and reverses its radial motion.
02 · Properties
Key properties
- Integer k = R/r → k cusps and closure after one revolution.
- One arch has length 8r(R+r)/R; total length 8(R+r) for integer k.
- Enclosed area: π(R+r)(R+2r).
- k = 1 is the cardioid; k = 2 is the nephroid.
- Rational k = p/q closes after q revolutions; irrational k never closes.
03 · Questions
Epicycloid: common questions
- What is the equation of the Epicycloid?
- Parametric equations: x = (R+r) cos t - r cos(((R+r)/r)t), y = (R+r) sin t - r sin(((R+r)/r)t). R is the fixed circle’s radius (R = 1 in the graph), r the rolling circle’s, and k = R/r.
- What is the perimeter of the Epicycloid?
- Perimeter (whole-number k): L = 8(R + r). R is the fixed circle’s radius (R = 1 in the graph), r the rolling circle’s, and k = R/r.
- What is the area of the Epicycloid?
- Area: A = π(R + r)(R + 2r). R is the fixed circle’s radius (R = 1 in the graph), r the rolling circle’s, and k = R/r.
04 · Wonders
See it in a math wonder
05 · Keep exploring