Polar Coordinates Calculator
Convert a point between Cartesian (x, y) and polar (r, θ) coordinates, with the angle in the right quadrant, in degrees and radians.
Result
Point: (5, 53.1301023542°)
Exact conversion; decimals shown to six places.
- r, distance from the origin
- 5
- θ, from −180° to 180°
- 53.130102 ° (0.927295 rad)
- θ, from 0° to 360°
- 53.130102 ° (0.927295 rad)
- Where it lies
- Quadrant I
https://calcgallery.com/geometry/polar-coordinates-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Polar Coordinates Calculator
- Choose which way to convert.
- For Cartesian to polar, enter x and y. For polar to Cartesian, enter r and the angle θ, in degrees or radians.
- Read the converted point, and the angle both from −180° to 180° and from 0° to 360°.
The conversions
to polar r = √(x² + y²) θ = atan2(y, x)to Cartesian x = r cos θ y = r sin θOne point, many names
A point has one pair of Cartesian coordinates but many polar ones: adding 360° to θ, or turning 180° and making r negative, reaches the same spot. (5, 53.13°), (5, 413.13°) and (−5, 233.13°) are all the point (3, 4). Going to polar, the page gives the usual answer, a positive r with θ in either of the two standard ranges; going to Cartesian, any r and θ are accepted, negative r included.
The origin
At (0, 0) the distance is 0 and there is no direction to measure, so every angle describes the point. The page gives r = 0 and says so instead of choosing an angle. For the distance between two points rather than from the origin, use the distance calculator; for the sine and cosine themselves, the trigonometry calculator.
With your numbers
- r = √(3² + 4²) = 5
- θ = atan2(4, 3) = 53.130102°
Quick reference
Exact conversion; decimals shown to six places.
| Example | Point |
|---|---|
| (3, 4) | (5, 53.1301023542°) |
| (−3, 4) | (5, 126.869897646°) |
| (−1, −1) | (1.41421356237, -135°) |
| (0, −2) | (2, -90°) |
| r 2 at 90° | (0, 2) |
| r 4 at 150° | (-3.46410161514, 2) |
| r 5 at 53.13° | (3.00000000029, 3.99999999978) |
Common questions about the Polar Coordinates Calculator
- How do I convert (x, y) to polar coordinates?
- r is the distance from the origin, √(x² + y²), and θ is the angle from the positive x-axis to the point. For (3, 4), r = 5 and θ ≈ 53.13°. The angle must be taken in the quadrant where the point actually is, which the next answer explains.
- Why is arctan(y/x) sometimes wrong?
- Because y/x loses the signs. (3, 4) and (−3, −4) give the same ratio, 4/3, so arctan gives 53.13° for both, but the second point is in the third quadrant at −126.87°. The page uses the two-argument arctangent, which looks at the signs of x and y separately and always lands in the right quadrant.
- How do I convert polar coordinates to (x, y)?
- x = r cos θ and y = r sin θ. A radius of 4 at 150° gives x = 4 cos 150° ≈ −3.4641 and y = 4 sin 150° = 2.
- Which angle range should I use?
- Either; they name the same direction. −135° and 225° are the same line from the origin, turned the other way round. Many textbooks give θ from 0° to 360° (or 0 to 2π); programming languages and the atan2 function give −180° to 180° (−π to π). The page shows both.