What is polar coordinate?

In the $xy$ -plane, a point can be described by two polar variables:

So we write a point as $(r,\theta)$ , and geometrically it is determined by a length and a direction from the origin.

r =θ (rad) =

Cartesian point: (2.79, 2.87)

Polar coordinate: (4.00, 0.80)

💡 Tip: Drag the blue point or click anywhere on the graph to choose a point. The Cartesian and polar coordinates are computed automatically.

In this note, we use the convention:

When $r$ changes to $-r$ , the same point is obtained by adding $\pi$ to the angle:

(-r,\theta) \equiv (r,\theta + \pi).

In the graph below, when $r$ is negative, the vector is drawn as a dashed segment.

r =θ (rad) =

Cartesian point: (-3.06, -2.58)

Polar coordinate: (-4.00, 0.70)

Since r < 0, this is equivalent to (4.00, 3.84).

💡 Tip: Drag the blue point or click anywhere on the graph to choose a point. The Cartesian and polar coordinates are computed automatically.

Relation with the Cartesian coordinate

Given $(r,\theta)$ , Cartesian coordinates are:

x = r\cos\theta, \quad y = r\sin\theta.

Given $(x,y)$ , polar variables are:

r = \sqrt{x^2 + y^2}, \quad \theta = \operatorname{arctan}(y/x) \; (\text{then normalized to } \theta \ge 0).

The interactive component below supports both input styles:

Polar mode: input $(r,\theta)$ and observe the point, radius, and right triangle with dotted sides.
Cartesian mode: input $(x,y)$ and observe the same geometric triangle and converted polar values.
Switch mode: use the button to move between polar and Cartesian input.

r =θ (rad) =

Cartesian point: (3.98, 3.03)

Polar coordinate: (5.00, 0.65)

💡 Tip: Drag the blue point or click anywhere on the graph to choose a point. The Cartesian and polar coordinates are computed automatically.

Polar coordinates are especially helpful in situations with radial symmetry or angular structure. Some advantages are:

Natural for circles and radial shapes: many equations become simpler than in Cartesian form.
Clear geometric meaning: $r$ is distance from the origin and $\theta$ is direction.
Easier description of closed curves: many closed curves can be parametrized directly by $\theta$ .

For example, the closed curve

r = 2 + \sin(3\theta), \quad 0 \le \theta \le 2\pi

is compact in polar form. If we convert this into a parametric equation in the $xy$ -coordinate system, we get the precise formulas

x(t) = \bigl(2 + \sin(3t)\bigr)\cos t, \quad y(t) = \bigl(2 + \sin(3t)\bigr)\sin t,

where $t=\theta$ and $0 \le t \le 2\pi$ .