What is the equation of the Cissoid of Diocles?

Polar equation: r = 2a sin θ tan θ; Cartesian equation: y^2(2a - x) = x^3. a is the radius of the generating circle; the curve approaches the line x = 2a.

What is the area of the Cissoid of Diocles?

Area (between the curve and its asymptote): A = 3π a^2. a is the radius of the generating circle; the curve approaches the line x = 2a.

Polar curves

Cissoid of Diocles

The cusped curve r = 2a·sin θ·tan θ invented around 180 BC to solve the ancient problem of doubling the cube.

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a (asymptote at x = 2a)1

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Formulas and key values

Polar equation: $r = 2a\sin\theta\tan\theta$
Cartesian equation: $y^2(2a - x) = x^3$
Asymptote: $x = 2a$
Area · between the curve and its asymptote: $A = 3\pi a^2$

a is the radius of the generating circle; the curve approaches the line x = 2a.

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01 · Definition

What is the Cissoid of Diocles?

The cissoid of Diocles — from the Greek for “ivy-shaped” — was constructed around 180 BC as a tool for doubling the cube: finding a cube with exactly twice the volume of a given one. Compass and straightedge cannot do it, but intersecting this curve with a line yields the required ∛2 ratio exactly.

The curve has a cusp at the origin and hugs the vertical asymptote x = 2a. Geometrically it is generated from a circle of diameter 2a: for each ray from the origin, the cissoid point copies the distance between where the ray leaves the circle and where it meets the tangent line at the far end.

02 · Properties

Key properties

Cusp at the origin; vertical asymptote at x = 2a.
The area between the curve and its asymptote is 3πa².
Cartesian form: y²(2a − x) = x³, a cubic curve.
It is the pedal-related cissoid of a circle and its tangent line, taken from the point opposite the tangency.
Newton later showed how to draw it with two rulers and a fixed right angle.

03 · Questions