Hypocycloid
The star traced by a circle rolling inside a fixed circle; k = 3 gives the deltoid, k = 4 the astroid.
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Formulas and key values
- Parametric equations
- Cusps · whole-number kIn the graph = 5
- Perimeter · whole-number kIn the graph = 6.4
- AreaIn the graph ≈ 1.508
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 Hypocycloid?
Let a circle roll without slipping around the inside of a larger circle: a point on its rim traces the hypocycloid, a star with k = R/r inward-curving sides when the ratio is an integer. The family contains famous members — k = 3 is the deltoid, k = 4 the astroid.
The case k = 2 is a genuine surprise: the traced “curve” degenerates to a straight diameter. This Tusi couple, described by Nasir al-Din al-Tusi in 1247, converts rotation into exact straight-line motion and reappeared centuries later in Copernican astronomy and in mechanical linkages.
02 · Properties
Key properties
- Integer k = R/r → k cusps; the curve stays inside the fixed circle.
- k = 2 degenerates to a straight diameter (the Tusi couple).
- One arch has length 8r(R−r)/R; enclosed area is π(R−r)(R−2r).
- k = 3 is the deltoid; k = 4 is the astroid.
- The cusps touch the fixed circle where the tracing point meets it.
03 · Questions
Hypocycloid: common questions
- What is the equation of the Hypocycloid?
- 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 Hypocycloid?
- 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 Hypocycloid?
- 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 · Keep exploring