Cycloid
The arch traced by a point on a rolling wheel — solution of both the brachistochrone and tautochrone problems.
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Formulas and key values
- Parametric equations
- Arc length · one archIn the graph = 8
- Area · under one archIn the graph ≈ 9.425
- Arch width
r is the radius of the rolling circle (r = 1 in the graph) and t is the angle it has turned.
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Start the video01 · Definition
What is the Cycloid?
Mark a point on the rim of a wheel and roll the wheel along a straight line: the point rises and falls in a chain of arches called the cycloid. Each arch spans 2πr horizontally and reaches height 2r, with a cusp where the point touches the ground. Galileo named the curve and tried to measure its area by weighing cut-outs of it.
The cycloid earned the nickname “the Helen of geometers” for the quarrels it caused. It answers two famous physics questions at once: an upside-down cycloid is the brachistochrone (the fastest slide between two points under gravity, proved via the calculus of variations after Johann Bernoulli’s 1696 challenge) and the tautochrone (from any starting point, a bead reaches the bottom in the same time — Huygens’ route to an ideal pendulum clock).
02 · Properties
Key properties
- One arch has length 8r — exactly four wheel diameters (Wren, 1658).
- The area under one arch is 3πr² — three times the rolling circle’s area.
- Cusps occur at every ground contact, where speed is momentarily zero.
- Inverted, it is both the brachistochrone and the tautochrone.
- Its evolute is another cycloid of the same size, shifted half an arch.
03 · Questions
Cycloid: common questions
- What is the equation of the Cycloid?
- Parametric equations: x = r(t - sin t), y = r(1 - cos t). r is the radius of the rolling circle (r = 1 in the graph) and t is the angle it has turned.
- What is the arc length of the Cycloid?
- Arc length (one arch): L = 8r. r is the radius of the rolling circle (r = 1 in the graph) and t is the angle it has turned.
- What is the area of the Cycloid?
- Area (under one arch): A = 3π r^2. r is the radius of the rolling circle (r = 1 in the graph) and t is the angle it has turned.
04 · Keep exploring