What is the equation of the Folium of Descartes?

Cartesian equation: x^3 + y^3 = 3axy; Parametric equations: x = 3at/(1+t^3), y = (3at^2)/(1+t^3). a sets the size; the tip of the loop is at (3a/2, 3a/2).

What is the area of the Folium of Descartes?

Area (inside the loop): A = (3/2)a^2. a sets the size; the tip of the loop is at (3a/2, 3a/2).

Classic Cartesian curves

Folium of Descartes

The looped cubic x³ + y³ = 3axy that sparked the Descartes–Fermat duel over tangent lines.

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Formulas and key values

Cartesian equation: $x^3 + y^3 = 3axy$
Parametric equations: $x = \frac{3at}{1+t^3}$ $y = \frac{3at^2}{1+t^3}$
Asymptote: $x + y + a = 0$
Area · inside the loopIn the graph = 1.5: $A = \frac{3}{2}a^2$

a sets the size; the tip of the loop is at (3a/2, 3a/2).

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01 · Definition

What is the Folium of Descartes?

Descartes proposed the curve x³ + y³ = 3axy in 1638 and challenged Fermat to find its tangent lines — expecting the problem to embarrass him. Fermat’s method of adequality handled it easily, an early landmark on the road to derivatives. The name folium is Latin for “leaf”: the curve loops through the origin in the first quadrant and runs off along two arms.

The folium is the standard first example of implicit differentiation with a genuine payoff: the loop closes at a node where the curve crosses itself with two distinct tangents, and both arms approach the same asymptote x + y + a = 0. Its symmetry axis is the line y = x, where the loop peaks at (3a/2, 3a/2).

02 · Properties

Key properties

Node at the origin with tangents along both axes.
Asymptote: x + y + a = 0 for both arms.
The loop’s area is 3a²/2 — and equals the area between the arms and the asymptote.
Symmetric about y = x; the loop’s vertex is (3a/2, 3a/2).
Rational parametrization x = 3at/(1+t³), y = 3at²/(1+t³) covers the whole curve.

03 · Questions