Cartesian to polar coordinates

The formula

• r = √(x² + y²)
• v = atan2(y, x)

r is the distance from the origin, Pythagoras on the two coordinates. v is the angle from the positive x-axis, counter-clockwise, found from both signs by atan2, or from arctan(y / x) with a correction for the quadrant. The card above computes r; the coordinate converter gives the angle too.

A worked example

(3, 4): r = √(9 + 16) = 5, and v = arctan(4 / 3) = 53.13°, since x > 0 and no correction is needed. (−3, −4) has the same r = 5 and the same ratio y / x = 4 / 3, but it lies in the third quadrant: v = 53.13° + 180° = 233.13°.

Why arctan alone is not enough

arctan(y / x) returns an angle between −90° and 90°, so it covers the right half of the plane only; (−3, −4) and (3, 4) give the same answer. The rule: when x > 0 use arctan as it is; when x < 0 add 180°; when x = 0 the angle is 90° or 270° by the sign of y. atan2(y, x) does all of that in one call, and every calculator and programming language has it, though some name the arguments the other way round.

Writing the angle

Conventions differ. 233.13° can also be written −126.87°, and in radians 4.07; a bearing from north is 90° − v. State which one a result uses.

Common mistakes

• Taking arctan without looking at the signs. The left half of the plane lands in the right half.
• Dividing x by y. The ratio is y over x, rise over run.
• Forgetting the square root. x² + y² is r², not r.

Frequently asked questions