# What are the polar coordinates?

Converts a point between rectangular coordinates (x, y) and polar coordinates (r, θ), with θ in degrees and radians, and draws the point.

- Page: https://www.acalculator.org/math/polar-coordinates-calculator
- JSON spec: https://www.acalculator.org/math/polar-coordinates-calculator.json
- Version: 217cdc1300fc

## Default answer

Example with the default inputs (Convert (x, y) to polar, x 3, y 3): The point converts to (r, θ) = (4.242640687, 45°).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| to | Convert | Which way to convert: from (x, y) to (r, θ), or from (r, θ) to (x, y). |
| x | x | The horizontal coordinate of the point. |
| y | y | The vertical coordinate of the point. |
| r | r | The distance from the origin; a negative r points the opposite way. |
| t | θ | The angle from the positive x-axis, counterclockwise. |
| u | Angle unit | Whether the angle θ you type is in degrees or radians. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| point | Converted point | The point in the other form, each number to 10 significant figures. |
| x | x | The horizontal coordinate, r cos θ. |
| y | y | The vertical coordinate, r sin θ. |
| r | r | The distance from the origin, √(x² + y²). |
| degrees | θ in degrees | The angle from the positive x-axis, from 0° up to (not including) 360°. |
| radians | θ in radians | The same angle in radians, from 0 up to (not including) 2π. |

## Method

Rectangular to polar: r = √(x² + y²), θ = atan2(y, x), plus 360° when negative. Polar to rectangular: x = r cos θ, y = r sin θ.

## Assumptions

- θ is measured counterclockwise from the positive x-axis. The page gives θ from 0° up to 360°; adding or taking away any whole number of turns (360° or 2π) names the same point.
- A typed r may be negative: the point is then on the opposite side of the origin, as in OpenStax.

## Worked examples

1. to = toPolar, x = 3, y = 3 gives r = 4.242641, degrees = 45, radians = 0.785398, point = (r, θ) = (4.242640687, 45°). Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (Example 5: r = 3√2, θ = π/4).
2. to = toRect, r = 3, t = 90, u = deg gives x = 0, y = 3, point = (x, y) = (0, 3). Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (Example 3: (3, π/2) is (0, 3)).
3. to = toRect, r = -2, t = 0, u = rad gives x = -2, y = 0. Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (Example 4: (−2, 0) is (−2, 0)).
4. to = toPolar, x = -1, y = -1.732051 gives r = 2, degrees = 240, radians = 4.18879. Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates.
5. to = toRect, r = 2, t = 30, u = deg gives x = 1.732051, y = 1. Source: OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates, https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates.

## FAQ

### How do I convert rectangular coordinates to polar?

Find the distance r = √(x² + y²) and the angle θ whose tangent is y ÷ x, in the right quadrant. For (3, 3), r = √18 = 4.2426 and θ = 45° (π/4 radians).

### How do I convert polar coordinates to rectangular?

Multiply r by the cosine and the sine of the angle: x = r cos θ and y = r sin θ. The point (3, 90°) is x = 3 × 0 = 0, y = 3 × 1 = 3, so (0, 3).

### Why is my angle 240° and not 60°?

tan θ = y ÷ x is the same for a point and for the point opposite it through the origin. The page uses atan2, which looks at the signs of x and y, so (−1, −√3) gets 240° (third quadrant), not 60°.

### What range does θ use?

The page gives θ from 0° up to, but not including, 360° (0 to 2π radians). The same point also has angles θ ± 360°, and θ ± 180° with a negative r.

### Can r be negative?

Yes, when you convert from polar. A negative r points in the direction opposite to θ, so (−2, 0) is the point (−2, 0). When you convert to polar the page always gives r ≥ 0.

### What about the origin?

At (0, 0) the distance r is 0 and every angle names the same point, so θ has no single value. The page asks you to move the point away from the origin.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §10.3 Polar Coordinates: x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x; Examples 3 to 5. https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-3-polar-coordinates (retrieved 2026-10-02)