Coordinates Converter — Cartesian ↔ Polar
Convert between Cartesian coordinates (x, y) and polar coordinates (r, θ) in both directions. Useful for physics, engineering, and trigonometry problems where one representation is more natural than the other.
Conversion direction
- 1
Squared sum x² + y²
3² + 4² = 25 - 2
Angle θ = atan2(y, x)
atan2(4, 3) = 0.927295 - 3
θ in degrees
0.927295 × 180 ÷ π = 53.1301 - 4
Radius r = √(x² + y²)
√25 = 5
How does this calculator work?
Two representations, same point. Cartesian (x,y) → Polar: r = √(x²+y²), θ = atan2(y,x). Polar (r,θ) → Cartesian: x = r·cosθ, y = r·sinθ. This converter handles both directions and returns the angle in degrees and radians with a vector diagram.
Formula
How this is calculated
Any point in the 2-D plane can be described in two equivalent ways. In Cartesian (rectangular) coordinates, a point is located by its horizontal distance x from the origin and vertical distance y. In polar coordinates, it is located by its distance r from the origin (the radius) and the angle θ it makes with the positive x-axis measured counter-clockwise.
To convert Cartesian to polar: r = √(x² + y²) gives the radius (always non-negative), and θ = atan2(y, x) gives the angle in the correct quadrant. The atan2 function (as opposed to plain arctan) handles all four quadrants correctly and avoids division-by-zero for vertical points. The angle is reported in both degrees and radians.
To convert polar to Cartesian: x = r·cos(θ) and y = r·sin(θ), where θ is supplied in degrees and converted internally. Negative r is undefined for the standard polar coordinate system, so this calculator requires r ≥ 0. Note that polar coordinates are not unique — adding any multiple of 360° to θ gives the same point.
Frequently asked questions
r = √(x²+y²) for the radius. θ = atan2(y, x) for the angle (in radians; multiply by 180/π for degrees). Use atan2 not arctan(y/x) to get the correct quadrant automatically.
x = r·cos(θ) and y = r·sin(θ), where θ must be in radians for the trig functions (or convert from degrees first by multiplying by π/180).
No. The same point can be written with any angle θ + 360°·n for any integer n, and r = 0 maps to the origin for any θ. The standard representation uses 0 ≤ θ < 360° (or −180° < θ ≤ 180°) and r ≥ 0.
Also known as
TG we-Calculate Editorial Team. (2026). Coordinates Converter — Cartesian ↔ Polar [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/coordinates-converter
TG we-Calculate Editorial Team. "Coordinates Converter — Cartesian ↔ Polar." TG we-Calculate. 2026. https://we-calculate.com/calculator/coordinates-converter.
TG we-Calculate Editorial Team, "Coordinates Converter — Cartesian ↔ Polar," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/coordinates-converter
@misc{wecalculate_coordinates_converter, title = {Coordinates Converter — Cartesian ↔ Polar}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/coordinates-converter}}, year = {2026}, note = {TG we-Calculate} }
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