Polar Coordinates Formula

Polar coordinates are a coordinate system where each point in the plane is described by a distance r from the origin and an angle θ from the positive x-axis, written as (r, θ).

The Formula

x=rcos⁡θ,y=rsin⁡θ
r=x2+y2,θ=arctan⁡ ⁣(yx)

When to use: Instead of 'go right 3, up 4' (Cartesian), polar says 'go 5 units in the direction of 53°.' It's how a radar works—distance and direction from a central point. Some shapes that look complicated in Cartesian coordinates become beautifully simple in polar.

Quick Example

The Cartesian point (1,1) in polar:
r=12+12=2,θ=arctan⁡ ⁣(11)=π4
So (1,1)=(2, π4) in polar.

Notation

A point is written (r,θ). By convention, r≥0 and θ∈[0,2π) or (−π,π], though negative r is sometimes allowed (meaning go in the opposite direction).

What This Formula Means

A coordinate system where each point in the plane is described by a distance r from the origin and an angle θ from the positive x-axis, written as (r,θ).

Instead of 'go right 3, up 4' (Cartesian), polar says 'go 5 units in the direction of 53°.' It's how a radar works—distance and direction from a central point. Some shapes that look complicated in Cartesian coordinates become beautifully simple in polar.

Formal View

(r,θ)↦(x,y)=(rcos⁡θ, rsin⁡θ); inverse: r=x2+y2, θ=atan2(y,x)

Worked Examples

Example 1

easy
Convert the polar coordinates (4,π3) to rectangular (Cartesian) coordinates.

Answer

(2,23)

First step

1
Use the conversion formulas: x=rcos⁡θ and y=rsin⁡θ.

Full solution

  1. 2
    x=4cos⁡(π3)=4⋅12=2.
  2. 3
    y=4sin⁡(π3)=4⋅32=23.
Polar coordinates (r,θ) locate a point by its distance from the origin and angle from the positive x-axis. Converting to rectangular uses x=rcos⁡θ and y=rsin⁡θ, which come from the right triangle formed by the point, the origin, and the projection onto the x-axis.

Example 2

medium
Convert the rectangular point (−3,3) to polar coordinates with r>0 and 0≤θ<2π.

Example 3

medium
Convert the rectangular point (−2,−23) to polar coordinates with r>0 and 0≤θ<2π.

Common Mistakes

  • Reading (r,θ) as (x,y) - the first number is a distance, the second an angle.
  • Mishandling the angle's quadrant in θ=arctan⁡(y/x) - arctan alone can land in the wrong quadrant; check the signs of x and y.
  • Forgetting a point has many polar names - adding 2π to θ (or negating r and adding π) gives the same point.

Why This Formula Matters

Radar, navigation, and circular/rotational motion are all distance-and-direction problems where polar is the native language, and many curves (roses, spirals) become one-line equations. The conversion formulas x=rcos⁡θ, y=rsin⁡θ are the bridge between this view and Cartesian. Recognizing it by "Is the location given as a distance from the origin plus an angle, rather than horizontal and vertical amounts?" — rather than by familiar numbers — is what lets a student tell it apart from cartesian coordinates and vectors (magnitude-direction form) and complex numbers (polar form) in a mixed problem set.

Frequently Asked Questions

What is the Polar Coordinates formula?

A coordinate system where each point in the plane is described by a distance r from the origin and an angle θ from the positive x-axis, written as (r,θ).

How do you use the Polar Coordinates formula?

Instead of 'go right 3, up 4' (Cartesian), polar says 'go 5 units in the direction of 53°.' It's how a radar works—distance and direction from a central point. Some shapes that look complicated in Cartesian coordinates become beautifully simple in polar.

What do the symbols mean in the Polar Coordinates formula?

A point is written (r,θ). By convention, r≥0 and θ∈[0,2π) or (−π,π], though negative r is sometimes allowed (meaning go in the opposite direction).

Why is the Polar Coordinates formula important in Math?

Radar, navigation, and circular/rotational motion are all distance-and-direction problems where polar is the native language, and many curves (roses, spirals) become one-line equations. The conversion formulas x=rcos⁡θ, y=rsin⁡θ are the bridge between this view and Cartesian. Recognizing it by "Is the location given as a distance from the origin plus an angle, rather than horizontal and vertical amounts?" — rather than by familiar numbers — is what lets a student tell it apart from cartesian coordinates and vectors (magnitude-direction form) and complex numbers (polar form) in a mixed problem set.

What do students get wrong about Polar Coordinates?

The procedure for polar coordinates is the easy part; the trap is reading (r,θ) as (x,y). Asking "Is the location given as a distance from the origin plus an angle, rather than horizontal and vertical amounts?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Polar Coordinates formula?

Before studying the Polar Coordinates formula, you should understand: trigonometric functions, unit circle, radian measure.

← Back to Polar Coordinates See All Examples Practice Problems