Polar ↔ Cartesian Coordinate Converter
Two common 2-D coordinate systems: Cartesian (x, y) and polar (r, θ). The tool converts in either direction live — enter (x, y) to get r = √(x² + y²) and θ = atan2(y, x); enter (r, θ) to get x = r·cos θ and y = r·sin θ. Toggle between degrees and radians at any time and see the equivalent angle in [0, 2π) plus the quadrant.
Please enter valid numbers (x, y or r, θ).
Cartesian (x, y)
3, 4
x = r·cos(θ), y = r·sin(θ)
Polar (r, θ)
r = 5, θ = 53.13°
r = √(x² + y²), θ = atan2(y, x)
Formula
Cartesian → polar: r = √(x² + y²), θ = atan2(y, x) Polar → Cartesian: x = r·cos θ, y = r·sin θ
- · Uses the standard math convention: the +x axis is 0° (or 0 rad) and angles increase counter-clockwise; 90° = π/2 rad points along +y.
- · θ follows the atan2 convention, returning (-180°, 180°] (or (-π, π] rad). The tool also reports the equivalent angle in [0°, 360°) (or [0, 2π)) so it lines up with the navigation / graphics convention.
- · At the origin (0, 0) the angle is undefined (atan2(0, 0) = 0); the tool reports 0 and labels the location as "origin".
- · A negative radius (r < 0) reflects the point through the origin — equivalent to r > 0 with θ + 180°. The tool applies r·cos θ and r·sin θ verbatim and does not auto-normalise.
- · Common pitfall: using atan(y/x) instead of atan2(y, x). The single-argument arctangent collapses quadrants II/III into IV/I and gives 180° errors — atan2 takes both signs into account and is what the tool uses internally.
- · Degree ↔ radian: 1 rad = 180/π ≈ 57.2958°; 1° = π/180 ≈ 0.01745 rad.
Frequently asked
When should I use Cartesian vs polar coordinates?
Cartesian (x, y) is the natural fit when horizontal and vertical motion are independent — robotics, CSS top/left positioning, CAD drafting, linear algebra. Polar (r, θ) is the natural fit for anything rotating about an origin — radar sweeps, planetary orbits (Kepler's 2nd law), polar plots (rose curves, cardioids), robot-arm joint angles, projectile / laser directions in games. Picking the right system can collapse a multi-line formula down to a single line.
Why does my calculator's atan(y/x) sometimes differ from this tool by 180°?
Because atan takes a single argument (the ratio y/x), it loses sign information and always returns an angle in (-90°, 90°]. For (-3, -4), y/x = 4/3 gives atan(4/3) ≈ 53.13° — but the point lies in quadrant III, so the correct angle is -126.87° (or 233.13°). The tool uses atan2(y, x), which inspects both signs to pick the quadrant: Q1 → 0–90°, Q2 → 90–180°, Q3 → -90 to -180°, Q4 → -90 to 0°, covering the full circle.
Degrees or radians — which should I pick?
For everyday and engineering use, degrees are more intuitive (360° = one revolution, 90° = right angle). For mathematical analysis, physics formulas (ω = 2π·f for circular motion, sin(ωt) for SHM, orbital mechanics) and most programming-language math libraries (Math.sin / Math.cos), use radians — there's no extra π/180 factor. The tool has a toggle, and switching units auto-converts the current angle so you don't have to retype it.
In computer graphics y points down — do I need to flip anything?
The tool uses the standard math convention (+y up, counter-clockwise positive). Most 2-D graphics frameworks (HTML canvas, CSS, SVG, SDL, Direct2D) put +y pointing down, so angles visually appear to grow clockwise. To map a math angle to screen space, negate θ (or negate y). 3-D engines that follow OpenGL / right-handed conventions (e.g. Unity) keep +y up and match this tool directly. The tool does not auto-flip the y-axis — apply the sign change yourself based on your engine.
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