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Astroid — Definition, Formula & Examples

An astroid is a star-shaped curve with four cusps, formed by rolling a circle of radius rr inside a circle of radius 4r4r. It belongs to the family of hypocycloids and appears frequently in parametric equations exercises.

The astroid is a hypocycloid of four cusps, defined as the locus of a point on the circumference of a circle of radius a/4a/4 rolling without slipping inside a fixed circle of radius aa. Its Cartesian equation is x2/3+y2/3=a2/3x^{2/3} + y^{2/3} = a^{2/3}, and it encloses an area of 38πa2\frac{3}{8}\pi a^2.

Key Formula

x=acos⁡3t,y=asin⁡3t,0≤t≤2πx = a\cos^3 t, \qquad y = a\sin^3 t, \qquad 0 \le t \le 2\pi
Where:
  • aa = Radius of the fixed outer circle, which determines the size of the astroid
  • tt = Parameter (angle), ranging from 0 to 2π to trace the full curve

How It Works

You typically encounter the astroid through its parametric form, which makes computing arc length, enclosed area, and tangent lines straightforward using standard calculus techniques. To sketch it, note that as tt goes from 00 to 2π2\pi, the curve traces four symmetric arcs meeting at cusps on the xx- and yy-axes at distance aa from the origin. The curve is symmetric about both axes and both lines y=±xy = \pm x.

Worked Example

Problem: Find the total arc length of the astroid with a=2a = 2.
Set up derivatives: Compute dx/dtdx/dt and dy/dtdy/dt from the parametric equations.
dxdt=−3acos⁡2tsin⁡t,dydt=3asin⁡2tcos⁡t\frac{dx}{dt} = -3a\cos^2 t\sin t, \qquad \frac{dy}{dt} = 3a\sin^2 t\cos t
Find the speed: Compute the integrand for arc length.
(dxdt)2+(dydt)2=3a∣cos⁡tsin⁡t∣=3a2∣sin⁡2t∣\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} = 3a|\cos t \sin t| = \frac{3a}{2}|\sin 2t|
Integrate using symmetry: By symmetry, the arc length in the first quadrant (00 to π/2\pi/2) is one-quarter of the total. Over this interval sin⁡2t≥0\sin 2t \ge 0.
L=4∫0π/23a2sin⁡2t dt=6a[−cos⁡2t2]0π/2=6aL = 4 \int_0^{\pi/2} \frac{3a}{2}\sin 2t\, dt = 6a\left[-\frac{\cos 2t}{2}\right]_0^{\pi/2} = 6a
Substitute a = 2: Plug in the given value of aa.
L=6(2)=12L = 6(2) = 12
Answer: The total arc length is 1212 units.

Why It Matters

The astroid is a standard example in calculus courses for practicing parametric arc length and area integrals because its trig-power parametrization leads to clean, closed-form answers. It also arises in mechanical engineering as the envelope of a family of line segments of fixed length sliding with endpoints on two perpendicular axes.

Common Mistakes

Mistake: Forgetting the absolute value in ∣sin⁡2t∣|\sin 2t| when computing arc length, leading to zero or a wrong answer over a full period.
Correction: The speed must always be non-negative. Use symmetry to integrate over an interval where sin⁡2t≥0\sin 2t \ge 0, then multiply accordingly.