Deltoid
The three-cusped hypocycloid, home of Steiner’s theorem: all Simson lines of a triangle envelope a deltoid.
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Formulas and key values
- Parametric equations
- PerimeterIn the graph = 16
- AreaIn the graph ≈ 6.283
- Cusps
r is the rolling circle’s radius; the fixed circle’s radius is 3r.
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Start the video01 · Definition
What is the Deltoid?
The deltoid — named for the Greek letter Δ it resembles — is the hypocycloid with three cusps, traced inside a circle three times the rolling radius. Euler met it in 1745 while studying optical caustics; Jakob Steiner’s 1856 study made it famous enough to also be called Steiner’s curve.
Steiner’s theorem is its showpiece: take any triangle, and for every point on its circumcircle draw the Simson line (the line through the feet of the perpendiculars to the three sides). All of these lines are tangent to a single deltoid. The curve also solved Kakeya’s needle question among convex-ish sets: a unit needle can be rotated 180° inside a deltoid of area π/8.
02 · Properties
Key properties
- Perimeter 16r and area 2πr² (twice the rolling circle).
- It is the hypocycloid with k = 3 (R = 3r).
- Any tangent chord between two branches has constant length 4r.
- Envelope of the Simson lines of any triangle (Steiner’s theorem).
- A unit segment can rotate fully inside a deltoid of area π/8.
03 · Questions
Deltoid: common questions
- What is the equation of the Deltoid?
- Parametric equations: x = r(2 cos t + cos 2t), y = r(2 sin t - sin 2t). r is the rolling circle’s radius; the fixed circle’s radius is 3r.
- What is the perimeter of the Deltoid?
- Perimeter: L = 16r. r is the rolling circle’s radius; the fixed circle’s radius is 3r.
- What is the area of the Deltoid?
- Area: A = 2π r^2. r is the rolling circle’s radius; the fixed circle’s radius is 3r.
04 · Keep exploring