Astroid
The four-cusped star x^(2/3) + y^(2/3) = a^(2/3), also the envelope of a sliding ladder.
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Formulas and key values
- Cartesian equation
- Parametric equations
- PerimeterIn the graph = 6
- AreaIn the graph ≈ 1.178
- Cusps
a is the distance from the center to each cusp (the fixed circle’s radius).
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Start the video01 · Definition
What is the Astroid?
The astroid is the four-cusped hypocycloid: roll a circle inside a circle four times its radius and a rim point draws the star x^(2/3) + y^(2/3) = a^(2/3). The exponent 2/3 makes it the Lamé curve (superellipse) with n = 2/3 — the concave extreme of that family.
It has a second, entirely different construction: slide a ladder of fixed length down a wall, and the region the ladder can never enter is bounded by an astroid. In other words, the astroid is the envelope of all segments of length a with endpoints on the two axes — a staple of envelope and related-rates problems.
02 · Properties
Key properties
- Perimeter 6a and area 3πa²/8 — both elementary despite the cusps.
- It is the hypocycloid with k = 4 (R = 4r).
- Envelope of a segment of length a sliding with ends on the axes.
- Every tangent line cuts the axes in a segment of constant length a.
- It is the evolute of an ellipse, stretched by the axis ratio.
03 · Questions
Astroid: common questions
- What is the equation of the Astroid?
- Cartesian equation: x^(2/3) + y^(2/3) = a^(2/3); Parametric equations: x = a cos^3 t, y = a sin^3 t. a is the distance from the center to each cusp (the fixed circle’s radius).
- What is the perimeter of the Astroid?
- Perimeter: L = 6a. a is the distance from the center to each cusp (the fixed circle’s radius).
- What is the area of the Astroid?
- Area: A = (3/8)π a^2. a is the distance from the center to each cusp (the fixed circle’s radius).
04 · Keep exploring