Polar ↔ Cartesian Coordinates Conversion Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
To convert from Cartesian to polar: r = √(x²+y²) and θ = atan2(y, x). To convert from polar to Cartesian: x = r×cos(θ) and y = r×sin(θ). The point (3, 4) in Cartesian corresponds to (5, 53.13°) in polar.
Explanation
A point in the plane can be located in two equivalent ways: in Cartesian coordinates (x, y), its horizontal and vertical position relative to an origin, or in polar coordinates (r, θ), its distance from the origin (r) and the angle (θ) the segment from the origin to the point makes with the horizontal axis. Both systems contain exactly the same information, simply organized differently, and some problems are more naturally solved in one system or the other: polar coordinates simplify, for example, the description of a circular motion or a wave, while Cartesian coordinates remain more intuitive for most common geometric uses. Converting from Cartesian to polar uses the Pythagorean theorem for the distance r (see also our hypotenuse calculator, which applies the same principle), and a two-argument arctangent function (atan2) for the angle, rather than a simple arctan(y/x): the latter function alone cannot distinguish a point in the first quadrant from a point in the third quadrant, both giving the same y/x ratio, whereas atan2 uses the signs of x and y separately to place the angle in the correct quadrant.
Example: convert the Cartesian point (3, 4) to polar
Inputs
Direction: Cartesian to polar. x = 3, y = 4.
Calculation
r = √(3² + 4²) = √25 = 5. θ = atan2(4, 3) ≈ 53.13°.
Result
The point (3, 4) in Cartesian coordinates corresponds to (5, 53.13°) in polar coordinates.
Frequently asked questions
Why use atan2 rather than simply arctan(y/x)?
Because the ratio y/x alone is identical for a point (3, 4) in the first quadrant and a point (−3, −4) in the third quadrant, which would make a simple arctan(y/x) unable to distinguish these two opposite directions. The atan2(y, x) function examines the signs of x and y separately, not just their ratio, which lets it correctly place the angle in the right quadrant, over the full range from −180° to 180°. This same quadrant distinction appears in calculating the angle between two vectors in our vector dot product calculator, although that one uses arccos rather than atan2.
What interval does the angle θ obtained fall in?
The angle θ calculated by atan2 falls between −180° and 180° (that is, between −π and π radians), a common convention that covers all possible directions around the origin without ambiguity. Some conventions prefer to express the angle between 0° and 360°: just add 360° to a negative result to get this equivalent representation.
What happens if the starting point is exactly the origin (0, 0)?
At the origin, the distance r is 0, but the angle θ then has no geometric meaning: a point exactly at the origin has no direction of its own, since it does not extend in any particular direction. The calculation stays numerically defined in this case (atan2(0,0) conventionally returns 0), but this result should not be interpreted as a real direction.