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Polar Coordinate System

Definition

The polar coordinate system specifies a given point in the plane R2\mathbb{R}^2 (see Cartesian coordinate system) by using two coordinates:

  • the point's distance from a reference point called the pole,
  • the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole.

The distance from the pole is called the radius, and the angle is called the polar angle.

Interpretation

The Cartesian product [0,∞)×R[0,\infty)\times\mathbb{R} supplies pairs (r,θ)(r,\theta); different angle values can represent the same point. The trigonometric functions convert the radius and direction to Cartesian coordinates through x=rcos⁡θx=r\cos\theta and y=rsin⁡θy=r\sin\theta.

The polar coordinate system locates a point on a plane by saying how far away it is from a centre point and what angle it makes from a fixed direction.

A polar grid of concentric circles and rays with a point P at distance r and angle theta, and its Cartesian coordinates r cos theta and r sin theta.

Using the 2-argument arctangent for the angle, for the nonzero point in the figure, r=x2+y2r=\sqrt{x^2+y^2} and θ=atan2⁡(y,x)\theta=\operatorname{atan2}(y,x). Constant rr gives a circle and constant θ\theta gives a ray of the polar grid.

Set the radius rr and the polar angle θ\theta, or drag the point: x=rcos⁡θx=r\cos\theta and y=rsin⁡θy=r\sin\theta.

Key Results

  • Conversion to Cartesian coordinates: x=rcos⁡θx = r \cos \theta, y=rsin⁡θ.y = r \sin \theta.
  • Conversion from Cartesian coordinates: r=x2+y2r = \sqrt{x^2 + y^2}, θ=atan2⁡(y,x).\theta = \operatorname{atan2}(y, x).

Examples

  • With r≥0r\ge0, θ=π/4\theta=\pi/4 describes the ray from the pole at angle π/4\pi/4 to the polar axis: (x,y)=(r/2,r/2)(x,y)=(r/\sqrt{2},r/\sqrt{2}).
  • r=1r = 1 is a unit circle centred at the pole.

Prerequisites

Sources

Aliases: polar coordinates Symbols: infinity, pi, real numbers, square root, theta

Last updated: 20 October 2026 03:02