Polar Coordinates Calculator

Convert between rectangular (x, y) and polar (r, θ) coordinates in either direction. Enter a point, pick degrees or radians, and get the conversion plus the quadrant and reference angle.

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How polar coordinate conversion works

Rectangular coordinates locate a point by walking: x steps right, y steps up. Polar coordinates locate the same point by aiming: face angle θ (measured counterclockwise from the positive x-axis) and walk r units straight ahead. Both name every point in the plane; which one is convenient depends on the problem. Circles, spirals, orbits, and anything rotating are painful in x and y but almost trivially clean in r and θ — the circle x² + y² = 25 becomes simply r = 5.

Going from polar to rectangular is pure trigonometry with no traps: x = r·cos θ, y = r·sin θ, done. The reverse direction has one famous trap, which is why this calculator uses atan2 — read on.

The formula

r = √(x² + y²)     θ = atan2(y, x)
x = r·cos θ     y = r·sin θ

Here (x, y) is the rectangular point, r is its distance from the origin (via the Pythagorean theorem), and θ is the angle from the positive x-axis. The function atan2(y, x) is the two-argument arctangent: it returns the angle of the point (x, y) anywhere in the plane, from −180° to +180°.

Worked example

Convert (3, 4) to polar. The radius is r = √(3² + 4²) = √25 = 5 (the classic 3-4-5 right triangle). The angle is θ = atan2(4, 3) ≈ 53.13° (about 0.9273 radians), in Quadrant I. So (3, 4) in polar form is (5, 53.13°). Run it in reverse to check: x = 5·cos(53.13°) ≈ 3 and y = 5·sin(53.13°) ≈ 4. Round trips like that are the fastest way to catch a conversion mistake.

Why atan2 beats plain arctan (the quadrant trap)

The naive formula θ = arctan(y⁄x) loses information the moment you divide: the points (3, 4) and (−3, −4) give the identical ratio 4⁄3, so arctan hands you the identical angle 53.13° for both — even though (−3, −4) sits in Quadrant III, a full 180° away. Ordinary arctan can only return angles between −90° and +90° (Quadrants I and IV), so every Quadrant II and III point comes out wrong, and x = 0 crashes the division outright. The fix is atan2(y, x), which keeps the two signs separate instead of collapsing them into a ratio: it knows (−3, −4) means “left and down” and correctly returns −126.87°. If you must use plain arctan by hand, the patch-up rule is: add 180° when x < 0, and let the sign of y break the tie on the axis. Or skip the ritual and use atan2 — every programming language has it, and it exists precisely because of this trap.

One convention worth knowing: a negative r is legal in polar coordinates and means “walk backwards” — the point (−5, 53.13°) is the same as (5, 233.13°), plotted in the direction opposite the angle. Textbooks use it for polar curves like r = cos(2θ) where the formula naturally dips negative. This calculator accepts negative r and does the sign bookkeeping for you; also remember polar names aren’t unique — adding 360° to θ leaves the point unchanged.

Frequently asked questions

How do I convert rectangular coordinates to polar?

Two formulas: r = √(x² + y²) gives the distance from the origin (straight from the Pythagorean theorem), and θ = atan2(y, x) gives the angle from the positive x-axis. For example, (3, 4) becomes r = √25 = 5 and θ ≈ 53.13°, so the polar form is (5, 53.13°). Use atan2 rather than plain arctan so the quadrant comes out right.

Why should I use atan2 instead of arctan(y/x)?

Because dividing y by x destroys the sign information that tells you which quadrant the point is in: (3, 4) and (−3, −4) give the same ratio, so arctan returns the same 53.13° for both, even though the second point is 180° away in Quadrant III. Plain arctan only covers Quadrants I and IV, and x = 0 breaks it entirely. atan2(y, x) takes the signs separately and returns the correct angle anywhere in the plane.

Can r be negative in polar coordinates?

Yes, by convention: a negative r means the point is plotted in the direction opposite the angle — (−5, 53.13°) is the same point as (5, 233.13°). It shows up naturally when graphing polar curves like r = cos(2θ), where the formula dips below zero. This calculator accepts negative r in polar-to-rectangular mode and handles the sign automatically.

What is a reference angle and why does it matter?

The reference angle is the acute angle (between 0° and 90°) that your angle makes with the x-axis. It matters because every trig value of any angle equals plus-or-minus the same trig value of its reference angle — so 126.87° in Quadrant II has reference angle 53.13°, and sin(126.87°) = sin(53.13°). It's the key trick for evaluating trig functions outside the first quadrant.

Is the polar form of a point unique?

No — one point has infinitely many polar names. Adding any multiple of 360° to θ names the same point, and the negative-r convention adds even more aliases: (5, 53.13°), (5, 413.13°), and (−5, 233.13°) are all identical. The standard choice, and what this calculator returns, is r ≥ 0 with θ between −180° and 180°. Rectangular coordinates, by contrast, are unique — one point, one (x, y).

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