Polar Coordinates Calculator
A point in the plane can be named two ways: Cartesian coordinates (x, y), measured along two perpendicular axes, or polar coordinates (r, θ), measured as a distance from the origin plus an angle from the positive x-axis. This calculator converts in both directions, resolving the angle with atan2 so the quadrant is always right, and reports θ in degrees and radians. It is a two-way coordinate converter with 6-decimal precision, not a grapher or a triangle solver.
θ is reported in the range 0° to 360° (0 to 2π radians), resolved to the correct quadrant with atan2. The origin (0, 0) returns r = 0 and θ = 0 by convention.
Quick Answer
For the point (x, y) = (3, 4): r = √(3² + 4²) = 5.000000, and θ = atan2(4, 3) = 53.130102° (0.927295 rad). Going the other way, (r, θ) = (5, 60°) gives x = 5 · cos 60° = 2.500000 and y = 5 · sin 60° = 4.330127. The calculator above has (3, 4) pre-filled; press Calculate to see all three values at once.
Key takeaways
- Polar coordinates name a point by distance from the origin r and angle θ from the positive x-axis; Cartesian coordinates use horizontal x and vertical y.
- The angle is computed with atan2(y, x), which puts θ in the correct quadrant automatically, something the plain arctan(y/x) formula cannot do.
- θ is reported in the range 0° to 360° (0 to 2π rad), so (3, −4) comes out as 306.869898°, never −53.130102°.
- A negative radius is rejected, not silently converted: represent the same point as (r, θ + 180°) instead.
- Results are shown to 6 decimal places and the results box stays hidden until you calculate.
How to use the polar coordinates calculator
Pick a direction, enter the numbers, and read the answer in both angle units at once.
- Choose the conversion direction: Rectangular to polar starts from x and y; Polar to rectangular starts from r and θ. The (3, 4) example is pre-filled in the first mode.
- Enter the values: decimals work everywhere. In polar mode, pick degrees or radians for θ beside the angle field; the default is degrees.
- Click Calculate: rectangular mode returns r plus θ in degrees and in radians; polar mode returns x and y. The results appear below the buttons.
- Copy or share: each results box has Copy my results and Share results side by side. Reset restores the worked-example values.
Switching direction clears the old results so a polar answer can never be mistaken for a Cartesian one. Validation messages name the exact problem: a missing number reports "Please enter valid x and y coordinates." (or the radius/angle equivalent), and a negative radius in polar mode reports "Radius must be non-negative."
The conversion formulas
Polar to Cartesian multiplies the radius by the cosine and sine of the angle; Cartesian to polar uses the Pythagorean distance for r and the two-argument arctangent for θ.
Cartesian → Polar: r = √(x² + y²), θ = atan2(y, x)
The plain formula θ = arctan(y/x) is where hand calculations and weaker tools go wrong: it cannot tell (3, 4) from (−3, −4), since both give the same ratio, and it fails outright at x = 0. The atan2 function used here takes y and x separately and returns the angle in the correct quadrant, which is the approach taught in standard calculus treatments of polar coordinates such as OpenStax Calculus Volume 2 and the worked notes at Paul's Online Math Notes. The trigonometric functions themselves, with their exact definitions, are documented in the NIST Digital Library of Mathematical Functions.
Worked example: (3, 4), converted both ways
The calculator is pre-filled with x = 3, y = 4. Here is the full arithmetic it performs.
Switch the tool to Polar to rectangular and it already holds r = 5, θ = 60° as a second check: x = 5 · cos 60° = 5 · 0.5 = 2.500000, y = 5 · sin 60° = 5 · 0.866025 = 4.330127. The table below is six more real runs of this calculator; every row reproduces exactly if you type the inputs in.
| Input | Output |
|---|---|
| (5, 60°) polar → rectangular | x = 2.500000, y = 4.330127 |
| (π/3 rad, r = 5) polar → rectangular | x = 2.500000, y = 4.330127 (same point, angle in radians) |
| (−3, 4) rectangular → polar | r = 5.000000, θ = 126.869898° = 2.214297 rad |
| (3, −4) rectangular → polar | r = 5.000000, θ = 306.869898° = 5.355890 rad |
| (0, 5) rectangular → polar | r = 5.000000, θ = 90.000000° = 1.570796 rad |
| (10, 180°) polar → rectangular | x = −10.000000, y = 0.000000 |
Quadrants, signs and why atan2 matters
The angle of a point depends on which quadrant it sits in, and this is the single most common source of wrong answers in hand conversions.
Take the four sign combinations of (3, 4). The plain arctan(4/3) gives 53.130102° for all four, which is wrong in three of them. The calculator's atan2-based angle is quadrant-correct for each:
| Point (x, y) | Quadrant | θ in degrees | θ in radians |
|---|---|---|---|
| (3, 4) | I | 53.130102° | 0.927295 |
| (−3, 4) | II | 126.869898° | 2.214297 |
| (−3, −4) | III | 233.130102° | 4.068888 |
| (3, −4) | IV | 306.869898° | 5.355890 |
All four values are real runs of the calculator above. The angle convention matters whenever polar points feed another computation: the polar distance calculator takes the angular difference between two polar points, so a quadrant-flipped angle there corrupts the distance. Under the hood the cosine and arccosine relationships are the same ones the cosine calculator and inverse cosine calculator work with.
What competing calculators miss
Five established tools cover this keyword, and each leaves something on the table:
- Omni Calculator's pair: a Cartesian-to-polar tool and a separate polar-to-Cartesian tool, each well explained. The gap is the split itself: converting round-trip means two pages, two input forms, and re-typing the numbers.
- BraveCalculator's polar coordinates page: both directions in one tool with a degree/radian choice, the closest rival to this page. Its guidance on the quadrant problem and on what happens at the origin is thin.
- FIRGELLI's engineering converter: polar to Cartesian only, degrees only, built for actuator positioning. Useful in its niche, but it cannot do the reverse conversion and never reports radians.
- NumberAnalytics and Study.com guides: thorough teaching material on the conversion formulas, including the atan2 caution. They are lessons, not calculators: no inputs, no answers.
- The gap this page fills: both directions in one tool, atan2 quadrant-correct angles reported in degrees AND radians in a single run, 6-decimal precision, a pre-filled worked example, and the honest edge cases (negative radius rejected, origin convention, 0°-to-360° normalization) stated on the page instead of failing silently.
Frequently asked questions
Polar coordinates describe a point as (r, θ), where r is the distance from the origin and θ is the angle measured counterclockwise from the positive x-axis. Cartesian coordinates describe the same point as (x, y), the horizontal and vertical distances along the axes.
Use x = r · cos(θ) and y = r · sin(θ). For (r, θ) = (5, 60°), x = 5 · cos 60° = 2.500000 and y = 5 · sin 60° = 4.330127. Choose degrees or radians for θ in the calculator before you convert.
Use r = √(x² + y²) and θ = atan2(y, x). For (x, y) = (3, 4), r = 5.000000 and θ = 53.130102° (0.927295 rad). The atan2 function places the angle in the correct quadrant automatically.
The angle is normalized to 0° to 360° (0 to 2π radians). A point like (3, −4) reports 306.869898°, not −53.130102°. Both name the same direction; this page uses the non-negative convention.
Not in this tool: a negative radius in polar-to-rectangular mode is rejected with "Radius must be non-negative." The point (−5, 60°) is the same as (5, 240°), so add 180° to θ and enter a positive r.
Yes. In polar-to-rectangular mode, pick Radians beside the angle field. Entering π/3 (about 1.047198) with r = 5 gives approximately (2.500000, 4.330127), the same point as 60°, up to the last decimal of the rounded input. Rectangular-to-polar mode always reports both degrees and radians, so no conversion is needed there.
The calculator returns r = 0.000000 and θ = 0.000000° by convention. Strictly, the angle of the origin is undefined because every direction from the origin reaches it, but 0 is the standard placeholder.
Angle conventions differ: some tools report atan2's raw range of −180° to 180°, while this page normalizes to 0° to 360°. For (3, −4) that is −53.130102° versus 306.869898°. Both name the same point, since the two differ by exactly one full turn.
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References
- OpenStax, Rice University. Calculus Volume 2: Parametric Equations and Polar Coordinates. Accessed October 23, 2026.
- Paul Dawkins, Lamar University. Paul's Online Math Notes: Polar Coordinates. Accessed October 23, 2026.
- Eric W. Weisstein. Polar Coordinates, MathWorld. Accessed October 23, 2026.
- National Institute of Standards and Technology. Digital Library of Mathematical Functions: Elementary Functions (Trigonometric). Accessed October 23, 2026.
- OpenStax, Rice University. Precalculus 2e: The Unit Circle and Trigonometric Functions. Accessed October 23, 2026.