Polar coordinates calculator
Rectangular to polar and back, with the right quadrant.
How do I convert rectangular coordinates to polar?
r = √(x² + y²) and tan θ = y ÷ x. Because tan θ repeats every 180°, the angle has to match the quadrant of the point: add 180° in Quadrants II and III, or use atan2(y, x), which does it for you.
How do I convert polar coordinates to rectangular?
x = r cos θ and y = r sin θ. For example, (3, π/2) becomes (3 × 0, 3 × 1) = (0, 3). A negative r points the opposite way: (−1, 2π/3) becomes (1/2, −√3/2).
Example: convert (−3, 4) to polar
r = √(9 + 16) = 5. The reference angle is arctan(4/3) = 53.13°, and the point is in Quadrant II, so θ = 180° − 53.13° = 126.87°, or 2.21 radians. The answer is (5, 2.21) — the same result OpenStax gets.
Why does my calculator give the wrong angle?
Plain arctan(y/x) only returns angles between −90° and 90°, so points on the left side of the plane come out in the wrong quadrant — (−1, −1) would look like 45° instead of 225°. This calculator uses the point’s quadrant, and shows both the 0°–360° angle and the −180° to 180° angle that atan2 returns.
Is a point’s polar form unique?
No. Adding 360° (2π) to θ gives the same point, and so does (−r, θ + 180°). At the origin r = 0 and any angle works.
Where do the formulas come from?
The conversion formulas and the worked examples are from OpenStax Precalculus 2e, section 8.3, and OpenStax Calculus Volume 2, section 7.3 (Polar Coordinates).
What angle do I get for a point in Quadrant III or IV?
For (−1, −1), the angle is 225° (or −135° measured the other way). For (3, −3) it is 315°, or −45°. Both forms point to the same location.
Reading the two forms
In rectangular form a point is the distance across (x) and up (y). In polar form it is the distance from the origin (r) and the angle θ measured counterclockwise from the positive x-axis. The two descriptions are linked by a right triangle with legs x and y and hypotenuse r, which gives x = r cos θ, y = r sin θ and r² = x² + y². Polar form suits circles, spirals and anything that rotates; rectangular form suits lines and grids.
Worked examples in both directions
From rectangular to polar, take (3, −3): r = √(9 + 9) = 4.243. The reference angle is arctan(3 ÷ 3) = 45°, and the point is in Quadrant IV, so θ = 360° − 45° = 315° (equivalently −45°). From polar to rectangular, take (6, 30°): x = 6 cos 30° = 5.196 and y = 6 sin 30° = 3. With a negative radius, (−2, 45°) lands at (−1.414, −1.414), directly opposite the point (2, 45°).
Common mistakes
Mixing degree and radian mode on a calculator is the classic error: cos 30 is 0.866 in degrees but 0.154 in radians. Taking arctan(y ÷ x) at face value is the next, since it ignores the quadrant; atan2 or a quadrant check fixes it. A third is dropping the sign of r. And at the origin the angle has no single value, because every direction reaches the same point, so the calculator reports an angle of 0 by convention.
Sources: OpenStax Precalculus 2e, section 8.3 (Polar Coordinates), and OpenStax Calculus Volume 2, section 7.3 (Polar Coordinates). Related: complex number calculator, trigonometry calculator, distance formula calculator and vector calculator.