Polar Coordinates Converter — Cartesian to polar, both ways

A polar coordinates converter moves a point in the plane between two ways of naming the same location. Cartesian coordinates give a point as (x, y): how far across and how far up. Polar coordinates give it as (r, θ): how far from the origin and in what direction. Neither is more correct — they suit different problems, and being able to move between them quickly is most of what the topic amounts to.

Arb Digital built this page because one specific step goes wrong constantly. Recovering the angle from x and y is not a single arctangent; it requires the quadrant, and the naive formula throws that information away. The tool uses the two-argument arctangent, shows which quadrant the point actually lands in, and draws the point so you can see whether the answer is plausible before you use it.

This Is Not a Map Coordinate Converter

Worth clearing up first, because the words overlap. Our coordinates converter deals with geographic positions on the Earth — latitude and longitude in decimal degrees, degrees-minutes-seconds, and UTM grid references. That is a problem about a curved surface, a datum and a projection.

This page is pure plane geometry. There is no Earth, no datum and no projection: just a flat plane, an origin, and a point described either by two distances or by one distance and one angle. If you arrived here looking to convert a GPS reading, the coordinates converter is the page you want. If you arrived here from a calculus, physics or complex-numbers problem, you are in the right place.

What This Converter Does

Choose a direction, enter the pair you have, and the tool fills in the other pair. It reports the converted point as the headline, and gives r, θ in degrees, θ in radians and the quadrant separately so nothing has to be re-derived. The angle-range control switches between the two standard conventions — 0 to 360° and −180 to 180° — because different textbooks, languages and instruments use different ones and the same point genuinely has both descriptions.

The breakdown rows add the reference angle, the exact angle in terms of π where one exists, and the equivalent complex-number form, since a + bi and r(cos θ + i sin θ) are the same conversion wearing different notation. Our complex number calculator works in that notation directly, and the vector calculator treats the same pair as a magnitude and direction.

How to Use It

1. Pick the direction. Cartesian to polar reads x and y; polar to Cartesian reads r and θ. The other two fields are outputs.
2. Choose degrees or radians before typing θ. Entering 1.5 meaning radians while the tool is in degrees gives an answer wrong by a factor of about 57.
3. Pick your angle range. Both conventions describe the same point; the tool shows both so you can quote whichever your course expects.
4. Check the quadrant against the drawing. If the point is drawn somewhere you did not expect, a sign is wrong in the input.
5. Read the π form when working in radians. Angles like π/4 and 2π/3 are recognised and printed exactly instead of as decimals.

The Formulas: How Each Direction Works

Polar to Cartesian is the easy direction and has no special cases at all: x = r·cos θ and y = r·sin θ. With r = 5 and θ = 53.1301°, that gives x = 5 × 0.6 = 3 and y = 5 × 0.8 = 4.

Cartesian to polar splits into two parts. The radius is Pythagoras: r = √(x² + y²), so the point (3, 4) sits at r = √(9 + 16) = √25 = 5. That part never causes trouble, because squaring removes the signs.

The angle is where the care is needed. The textbook formula θ = arctan(y ÷ x) is only correct when x is positive. Arctangent has a range of −90° to 90°, so it cannot return an angle in the second or third quadrant, and the ratio y ÷ x cannot distinguish (3, 4) from (−3, −4) — both give 4/3. The correct tool is the two-argument arctangent, written θ = atan2(y, x), which takes the signs of x and y separately and returns the angle in the right quadrant every time. The polar coordinate relations and the Jacobian that goes with them are set out in the DLMF section on the calculus of two or more variables.

Fixing the Angle by Hand Without atan2

If you only have a plain arctangent — on paper, or on a basic calculator — the correction is a rule with three cases. Compute the reference angle α = arctan(|y| ÷ |x|), which is always between 0° and 90°, then place it:

1. Quadrant I (x > 0, y > 0): θ = α.
2. Quadrant II (x < 0, y > 0): θ = 180° − α.
3. Quadrant III (x < 0, y < 0): θ = 180° + α.
4. Quadrant IV (x > 0, y < 0): θ = 360° − α, or equivalently −α.

Try it on (−1, −1). The reference angle is arctan(1) = 45°, the point is in quadrant III, so θ = 180° + 45° = 225°, and r = √2 ≈ 1.4142. A plain arctangent would have returned 45° and placed the point in quadrant I — diametrically opposite, and a mistake that survives every subsequent step of a calculation without ever looking wrong.

The axes are the remaining special cases. On the positive x-axis θ = 0°, on the positive y-axis θ = 90°, on the negative x-axis θ = 180°, and on the negative y-axis θ = 270° or −90°. At the origin itself, r = 0 and θ is genuinely undefined, because every direction is equally valid; this tool reports it as undefined rather than defaulting to zero.

The Same Point Has Infinitely Many Polar Names

This is the property that makes polar coordinates different in kind from Cartesian ones, and it is worth being explicit about. Adding a full turn to θ returns to the same point, so (5, 53.13°) and (5, 413.13°) and (5, −306.87°) are all the same location. Cartesian coordinates have no such freedom: one point, one (x, y) pair, always.

A negative r adds a second family. By convention r < 0 means "go the stated distance in the opposite direction", so (−5, 53.13°) is the same point as (5, 233.13°). Some courses forbid negative r entirely; others use it constantly when sketching polar curves, because it is what lets a curve like r = cos(2θ) trace all four petals of its rose. This tool accepts a negative r and normalises it, telling you what it did.

The consequence for anything automated is that you cannot test two polar points for equality by comparing their numbers. Convert both to Cartesian and compare there, or normalise the angle into a single agreed range first. The same trap appears when averaging angles: the mean of 350° and 10° is 180° if you average naively, and 0° if you do it properly.

Why Polar Coordinates Are Worth the Trouble

Any problem with a natural centre is simpler in polar form. A circle of radius 3 centred on the origin is x² + y² = 9 in Cartesian coordinates and simply r = 3 in polar — a constant. A spiral, a cardioid or a rose curve has a compact polar equation and an unpleasant Cartesian one. Anything involving rotation, orbits or radiation from a point source falls into this category.

In integration the payoff is larger still. Converting a double integral to polar coordinates replaces dx dy with r dr dθ, and that extra factor of r — the Jacobian of the transformation — is what makes integrals over discs and annuli tractable. It is also the trick behind evaluating the Gaussian integral, which has no elementary antiderivative in one dimension but falls out immediately in two once you switch to polar. MIT's Multivariable Calculus course covers that change of variables in detail.

The third use is in complex numbers. Multiplying two complex numbers in Cartesian form takes four real multiplications and some care with signs; in polar form you multiply the radii and add the angles, which is why powers and roots are so much easier there. De Moivre's theorem is that observation written down. Our trigonometric functions calculator and unit circle calculator cover the sine and cosine values that sit underneath all of it.

Common Mistakes to Avoid

• Using arctan(y ÷ x) alone. It cannot see the signs separately, so quadrants two and three come out wrong by exactly 180°.
• Mixing degrees and radians. Set the unit before typing the angle; a mismatch is an error of roughly a factor of 57.
• Assuming a point has one polar form. Adding 360° or negating r both give the same location, so equality must be tested after normalising.
• Forgetting the Jacobian in an integral. Converting to polar replaces dx dy with r dr dθ, and dropping the r changes the answer.
• Reading the origin as θ = 0. At r = 0 the angle is undefined, not zero, and treating it as zero can hide a bug in a rotation calculation.

Related Free Tools From Arb Digital

Use the coordinates converter for latitude, longitude and UTM, the vector calculator when the same pair is a magnitude and direction, the complex number calculator for the polar form of a + bi, and the angle converter to move between degrees and radians on their own. More live in the free tools hub.

Frequently Asked Questions

This converter is provided for study and for checking your own working. It is not a substitute for showing the quadrant reasoning your course or assessment requires.