Cartesian to polar converter

Type the point as two numbers, x then y: `(3, 4)`, `3, 4` or `3 4` all work. The polar form appears underneath, and Show the steps gives the working and draws the point with r and θ marked.

Set Angles to radians if your course works in them; the result then also shows the angle as a multiple of π. Angle range chooses between 0° to 360° and −180° to 180°, the two conventions you'll meet. To go back, press swap, or use the polar to Cartesian converter.

r is the distance from the origin, and Pythagoras gives it. For (3, 4), r = √(3² + 4²) = √25 = 5.

θ is the angle anticlockwise from the positive x axis. The textbook formula is θ = tan⁻¹(y/x), and it's right only when x is positive. For (−1, 1), y/x is −1 and tan⁻¹(−1) is −45°, which points into the fourth quadrant. The point is in the second, at 135°.

So the converter uses atan2(y, x), which looks at the signs of x and y separately and returns the angle in the right quadrant. The working says when the simple formula would have gone wrong.

If you're working by hand, find tan⁻¹(|y|/|x|) and place it. In the first quadrant (x and y positive) θ is that angle. In the second (x negative, y positive) take it from 180°. In the third (both negative) add it to 180°. In the fourth (x positive, y negative) take it from 360°.

On the axes there's nothing to calculate. Positive x is 0°, positive y 90°, negative x 180° and negative y 270°. At the origin r is 0 and any angle will do, so the converter gives 0.

atan2 first appeared in Fortran in 1961, taking y first and x second so that it matched the way the angle of a complex number is written. Almost every programming language since has kept that order.

Spreadsheets didn't. Excel's ATAN2 takes x first, as do OpenOffice Calc and Mathematica's ArcTan. Paste code-style arguments into a spreadsheet and the angle comes out reflected across the line y = x, a mistake that looks plausible because it's still a real angle.