Polar coordinates describe a point by its distance from the origin and its angle. They simplify circles, spirals, rotations, radar and navigation, and complex numbers in polar form.
How to use the Cartesian to Polar Coordinates Calculator
- Conversion
- x
- y
- r
- θ
Results update as you type, and everything is calculated in your browser.
How it works
The radius is the Pythagorean distance; the angle uses atan2 so that it lands in the right quadrant (−180° to 180°; add 360° for 0°–360°).
Inputs explained
| Field | What to enter |
|---|---|
| Conversion | Choose one: Cartesian → polar, Polar → Cartesian |
| x | A number |
| y | A number |
| r | A number |
| θ | Units: °, rad, π rad, grad, turn |
What each result means
- r = sqrt([x]^2 + [y]^2)
- θ = deg(atan2([y], [x]))
- θ (0° to 360°) = mod(deg(atan2([y], [x])), 360)
- x = [r]×cos(rad([θ]))
- y = [r]×sin(rad([θ]))
Example values
How the main result changes when x changes and everything else stays at the example values above:
| x | r |
|---|---|
| -1.5 | 4.272 |
| -2.25 | 4.5894 |
| -3 | 5 |
| -3.75 | 5.4829 |
| -4.5 | 6.0208 |
| -6 | 7.2111 |
Formula
r = √(x² + y²), θ = atan2(y, x)
x = r cos θ, y = r sin θ
Worked example
Inputs: Conversion: Cartesian → polar; x: -3; y: 4.
Results: r = 5; θ = 126.869898 °; θ (0° to 360°) = 126.869898 °.
Key terms
- Polar coordinates
- (r, θ): distance from the origin and angle.
Frequently asked questions
What is (−3, 4) in polar coordinates?
r = 5, θ ≈ 126.87°.
Why not just use arctan(y/x)?
It gives the wrong angle in quadrants II and III; atan2 fixes this.
Are polar coordinates unique?
No. Adding 360° to θ, or using −r with θ + 180°, gives the same point.
How are polar coordinates used with complex numbers?
a + bi = r(cos θ + i sin θ).