If the radius of the fixed circle is a then the equation is given by^[4] $${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}.}$$  This implies that an astroid is also a superellipse.

Parametric equations are $${\displaystyle {\begin{aligned}x=a\cos ^{3}t&={\frac {a}{4}}\left(3\cos \left(t\right)+\cos \left(3t\right)\right),\\[2ex]y=a\sin ^{3}t&={\frac {a}{4}}\left(3\sin \left(t\right)-\sin \left(3t\right)\right).\end{aligned}}}$$ 

The pedal equation with respect to the origin is $${\displaystyle r^{2}=a^{2}-3p^{2},}$$ 

the Whewell equation is $${\displaystyle s={3a \over 4}\cos 2\varphi ,}$$  and the Cesàro equation is $${\displaystyle R^{2}+4s^{2}={\frac {9a^{2}}{4}}.}$$ 

The polar equation is^[5] $${\displaystyle r={\frac {a}{\left(\cos ^{2/3}\theta +\sin ^{2/3}\theta \right)^{3/2}}}.}$$ 

The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation^[6] $${\displaystyle \left(x^{2}+y^{2}-a^{2}\right)^{3}+27a^{2}x^{2}y^{2}=0.}$$ 

The astroid is, therefore, a real algebraic curve of degree six.

The polynomial equation may be derived from Leibniz's equation by elementary algebra: $${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}.}$$ 

Cube both sides: $${\displaystyle {\begin{aligned}x^{6/3}+3x^{4/3}y^{2/3}+3x^{2/3}y^{4/3}+y^{6/3}&=a^{6/3}\\[1.5ex]x^{2}+3x^{2/3}y^{2/3}\left(x^{2/3}+y^{2/3}\right)+y^{2}&=a^{2}\\[1ex]x^{2}+y^{2}-a^{2}&=-3x^{2/3}y^{2/3}\left(x^{2/3}+y^{2/3}\right)\end{aligned}}}$$ 

Cube both sides again: $${\displaystyle \left(x^{2}+y^{2}-a^{2}\right)^{3}=-27x^{2}y^{2}\left(x^{2/3}+y^{2/3}\right)^{3}}$$ 

But since: $${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}\,}$$ 

It follows that $${\displaystyle \left(x^{2/3}+y^{2/3}\right)^{3}=a^{2}.}$$ 

Therefore: $${\displaystyle \left(x^{2}+y^{2}-a^{2}\right)^{3}=-27x^{2}y^{2}a^{2}}$$  or $${\displaystyle \left(x^{2}+y^{2}-a^{2}\right)^{3}+27x^{2}y^{2}a^{2}=0.}$$ 

Area enclosed^[7]

${\displaystyle {\frac {3}{8}}\pi a^{2}}$ 

Length of curve

${\displaystyle 6a}$ 

Volume of the surface of revolution of the enclose area about the x-axis.

${\displaystyle {\frac {32}{105}}\pi a^{3}}$ 

Area of surface of revolution about the x-axis

${\displaystyle {\frac {12}{5}}\pi a^{2}}$ 

The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities.

The dual curve to the astroid is the cruciform curve with equation ${\textstyle x^{2}y^{2}=x^{2}+y^{2}.}$  The evolute of an astroid is an astroid twice as large.

The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.^[8]

• Cardioid – an epicycloid with one cusp
• Nephroid – an epicycloid with two cusps
• Deltoid – a hypocycloid with three cusps
• Stoner–Wohlfarth astroid – a use of this curve in magnetics
• Spirograph

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 4–5, 34–35, 173–174. ISBN 0-486-60288-5.
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 10–11. ISBN 0-14-011813-6.
• R.C. Yates (1952). "Astroid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 1 ff.

• "Astroid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Astroid" at The MacTutor History of Mathematics archive
• "Astroid" at The Encyclopedia of Remarkable Mathematical Forms
• Article on 2dcurves.com
• Visual Dictionary Of Special Plane Curves, Xah Lee
• Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.