Polar graphing calculator

Type r = f(θ) and see the polar curve. Roses, cardioids, limaçons, spirals and lemniscates are all one line each, and sliders show how every parameter changes the shape.

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An eight-petal rose

r=3cos(4θ)

and a cardioid

r=1.5+1.5cos(θ)

How to use the polar graphing calculator

1
Type r = followed by any expression in theta (or θ), for example r = 1 + cos(theta).
2
θ runs from 0 to 12π by default, which closes almost every curve. Change it in the row’s settings for spirals or partial curves.
3
Add parameters: r = a cos(n theta) offers sliders for a and n.
4
Select the curve to see where it crosses the axes, refined to about twelve digits.

Polar coordinates in one paragraph

A polar equation gives the distance r from the origin as a function of the angle θ measured from the positive x-axis. Each (r, θ) becomes the point x = r cos θ, y = r sin θ. A negative r is allowed and puts the point on the opposite side of the origin, which is why some curves trace their loops twice. For more, see polar coordinates explained.

Rose curves: counting the petals

For r = a cos(nθ) or r = a sin(nθ) with a whole number n:

If n is odd, the rose has n petals. r = cos 3θ has three.
If n is even, it has 2n petals. r = cos 4θ has eight.
Each petal has length |a|.

Odd n gives n petals because the negative half of each cycle retraces petals that are already drawn. Fractional n, such as r = cos(2θ/5), gives overlapping roses that need a longer θ range to close.

The common polar families

Curve	Equation	Shape
Circle	`r = 2cos(theta)`	A circle through the origin
Cardioid	`r = 1 + cos(theta)`	A heart shape with a cusp
Limaçon	`r = 1 + 2cos(theta)`	A loop inside a loop
Rose	`r = cos(3theta)`	Petals
Archimedean spiral	`r = theta/4`	Equally spaced turns
Lemniscate	`r = 2sqrt(cos(2theta))`	A figure eight

Examples to try

Limaçon with an inner loop

r = 1 + 2cos(theta)

When the constant is smaller than the cosine coefficient, r goes negative and draws an inner loop.

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Archimedean spiral

r = theta/2

The distance between successive turns is constant: 2π × ½ = π.

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Lemniscate of Bernoulli

r = 2sqrt(cos(2theta))

A figure eight. r is undefined where cos 2θ < 0, which leaves the gaps between the lobes.

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Frequently asked questions

How do I type θ?

Type theta. It is shown as θ. The calculator also accepts the θ character if you paste it.

What range of θ is used?

From 0 to 12π by default, which closes almost every polar curve. You can set any other range in the row’s settings.

How many petals does r = cos(nθ) have?

n petals when n is odd, and 2n petals when n is even.

Can I convert a polar equation to Cartesian form?

Use x = r cos θ, y = r sin θ and r² = x² + y². For example r = 2cos θ becomes x² + y² = 2x, a circle of radius 1 centred at (1, 0).

Related tools and guides

Parametric equation grapher

Curves (x(t), y(t)) in 2D and 3D.

Polar coordinates explained

Rose curves

The cardioid

The heart curve

3D graphing calculator

Surfaces, implicit and parametric 3D graphs, exportable for 3D printing.

Try the polar graphing calculator

It opens instantly, with no account and nothing to install.

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