Cartesian to Polar Converter
Enter x and y to get polar coordinates (r, θ) with correct quadrant handling — r = √(x² + y²) and θ from atan2 — in degrees or radians.
Example: with x-coordinate 3 · y-coordinate 4 · Angle unit Degrees → Polar coordinates (r, θ): (5, 53.13°).
Computed by the calculator below using its default values. Change any input to see your own numbers.
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Check it outFrom (x, y) to (r, θ)
Polar coordinates describe the same point by distance and direction: r = √(x² + y²) and θ = atan2(y, x). For (3, 4): r = √(9 + 16) = 5 (the classic 3-4-5 triangle) and θ = tan⁻¹(4 ÷ 3) ≈ 53.13°. The quadrant is where plain tan⁻¹ goes wrong: for (−3, 4), tan⁻¹(4 ÷ −3) returns −53.13°, which points into the wrong quadrant entirely — atan2 keeps the signs of x and y separate and correctly reports 126.87°. Angles come out between −180° and 180°; add 360° when you need a positive angle, so (3, −4) is −53.13° or equivalently 306.87°.
Going back: polar to cartesian
The reverse conversion is x = r cos θ and y = r sin θ. Taking r = 5 at θ = 53.13°: x = 5 cos 53.13° ≈ 3 and y = 5 sin 53.13° ≈ 4, landing back on the original point. Keep your calculator’s angle mode matched to the units — 53.13 radians and 53.13 degrees are very different directions.
How it’s calculated
Radius r = √(x² + y²). Angle θ = atan2(y, x), the two-argument arctangent, which returns angles in (−180°, 180°] measured counterclockwise from the positive x-axis and handles all four quadrants plus x = 0 safely. Radians = degrees × π ÷ 180. Display rounding: r and radians to 4 decimals, degrees to 2. The inverse mapping is x = r cos θ, y = r sin θ.
Results update as you type and are estimates, not professional advice — verify important decisions with a qualified professional.
Common mistakes
- Using tan⁻¹(y ÷ x) blindly — in quadrants II and III it lands 180° off; use atan2 or add 180° whenever x < 0.
- Turning in a negative angle when the answer key wants 0–360°: add 360°, so −53.13° becomes 306.87°.
- Leaving a handheld calculator in radian mode while working in degrees (or vice versa).
Frequently asked questions
How do I convert cartesian coordinates to polar?
Compute r = √(x² + y²) and θ = atan2(y, x). For (3, 4): r = √25 = 5 and θ ≈ 53.13°, so the polar form is (5, 53.13°).
What happens when x is negative?
The point sits in quadrant II or III, where plain tan⁻¹(y ÷ x) is 180° off. atan2 fixes this automatically: (−3, 4) converts to (5, 126.87°), not (5, −53.13°).
How do I convert polar back to cartesian?
Use x = r cos θ and y = r sin θ. A point at r = 2, θ = 90° gives x = 2 cos 90° = 0 and y = 2 sin 90° = 2, i.e. (0, 2).
Should the angle be in degrees or radians?
Either works — they are two units for the same angle, related by 180° = π radians. This converter reports both: 53.13° is 0.9273 rad.
What is the polar form of the origin (0, 0)?
r = 0 with an arbitrary angle, since every direction gives the same point; the converter reports it as (0, 0°) by convention.