Cartesian to polar coordinates
Enter x and y and get the polar coordinates: the distance r to the origin and the angle θ, in degrees or radians, plus the quadrant the point lies in.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
The formulas
- r = √(x² + y²)
- θ = atan2(y, x)
atan2 takes the signs of x and y into account, so the angle is right in every quadrant. For (−1, −1), θ is −135° (or 225° measured from 0° to 360°), not 45°.
Polar coordinates
Instead of "3 right and 4 up", polar coordinates say "5 away at an angle of 53.13°". They are handy for anything that turns or radiates from a centre, such as radar, antennas and complex numbers. The other way round is the polar to Cartesian converter.
Worked examples
| Point (x, y) | r | θ |
|---|---|---|
| (3, 4) | 5 | 53.13° |
| (−1, −1) | 1.4142 | −135° (225°) |
| (0, −2) | 2 | −90° (270°) |
| (−3, 0) | 3 | 180° |
For (3, 4): r = √(9 + 16) = 5 and θ = atan2(4, 3) ≈ 53.13°. The point lies in quadrant I, which the result also states.
Complex numbers
A complex number a + bi is the point (a, b), so r is its modulus and θ its argument. Multiplying two complex numbers multiplies their r values and adds their angles, which is why polar form is the natural way to multiply and divide them. Try it with multiply complex numbers.
Going back
To check the result, convert it back with the polar to Cartesian converter: (5, 53.13°) gives x ≈ 3 and y ≈ 4. For the angle alone from a ratio, see the arctan calculator.
Frequently asked questions
Why use atan2 and not arctan(y/x)?
arctan(y/x) cannot tell (−1, −1) from (1, 1), because both give y/x = 1. atan2 looks at the signs of x and y, so (−1, −1) correctly gives −135° rather than 45°.
What happens at the origin?
A point at (0, 0) has r = 0 and no meaningful direction. The calculator reports it as the origin and shows an angle of 0°.
How do I get an angle between 0° and 360°?
Add 360° to a negative angle. The converter shows this value in its own row, so −135° appears as 225°.
Do I get degrees or radians?
Both are shown. The unit you choose above decides which one is in the headline answer.