Quick answer
To convert (x, y) to polar form, r = √(x² + y²) and θ = atan2(y, x). The point (3, 4) has r = 5 and θ = 53.13° (0.9273 rad).
Enter x and y. The calculator returns the radius and the angle in degrees and radians. It converts Cartesian coordinates into polar form.
Related tools: Polar to Rectangular Converter: x and y from radius and angle. Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results. Ratio Calculator: Simplify a ratio of two whole numbers.
How to use the Cartesian to Polar Converter
- Enter x.
- Enter y.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| x | A number | — |
| y | A number | — |
Formula and method
Formula
r = √(x² + y²); θ = atan2(y, x)
IEEE-754 double precision; shown to the decimals indicated.
Angle is in (−180°, 180°].
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: x = 3; y = 4.
| Result | Value |
|---|---|
| Radius r | 5 |
| Angle (degrees) | 53.130102 |
| Angle (radians) | 0.927295 |
Example 2
Inputs: x = -1; y = 1.
| Result | Value |
|---|---|
| Radius r | 1.414214 |
| Angle (degrees) | 135 |
| Angle (radians) | 2.356194 |
Example 3
Inputs: x = 2.5; y = 0.
| Result | Value |
|---|---|
| Radius r | 2.5 |
| Angle (degrees) | 0 |
| Angle (radians) | 0 |
The first line is the headline result (radius r). Below it the cartesian to Polar Converter lists angle (degrees) and angle (radians), which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Why atan2
atan(y ÷ x) cannot tell (−3, −4) from (3, 4), because both give the same ratio. atan2 uses the signs of both to place the angle in the right quadrant: (−3, −4) gives −126.87° (or 233.13°).
Going back
x = r cos θ and y = r sin θ. For r = 5 and θ = 53.13° that returns (3, 4), the check done on the Polar to Rectangular page.
How the result changes with x
Holding every other input at the first worked example, this table changes only x. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| x | Radius r | Angle (degrees) | Angle (radians) |
|---|---|---|---|
| 1.5 | 4.272002 | 69.443955 | 1.212026 |
| 3 | 5 | 53.130102 | 0.927295 |
| 6 | 7.211103 | 33.690068 | 0.588003 |
| 12 | 12.649111 | 18.434949 | 0.321751 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- x: left empty shows "Enter x." beside that field and no result.
- y: entering "z" shows "y must be a number." beside that field and no result.
Choosing between the Cartesian to Polar Converter and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Polar to Rectangular Converter](/math/polar-to-rectangular/) | radius r and angle θ | x |
| [Fraction Calculator](/math/fraction-calculator/) | numerator 1, denominator 1 and operation | Result |
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Random Number Generator](/math/random-number-generator/) | minimum, maximum and how many | Random numbers |
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
Assumptions and limitations
- Origin has an undefined angle (reported as 0)
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for x and y to confirm this.
Frequently asked questions
What is the polar form of (0, 0)?
r = 0 and the angle is undefined.
Are angles in degrees?
Both degrees and radians are shown.
What range does θ have?
The angle is reported from −180° to 180°; add 360° to a negative angle for 0° to 360°.
What does the Cartesian to Polar Converter calculate?
Radius and angle from x and y. It returns Radius r, Angle (degrees) and Angle (radians).
What formula does the Cartesian to Polar Converter use?
r = √(x² + y²); θ = atan2(y, x). Angle is in (−180°, 180°].
What do I need to enter in the cartesian to Polar Converter?
You enter x and y.
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Origin has an undefined angle (reported as 0).
What is an example of a cartesian to Polar result?
With x 3 and y 4, the radius r is 5.
Related calculators
- Polar to Rectangular Converter — x and y from radius and angle.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Random Number Generator — Random integers in a range, optionally repeatable with a seed.
- Area Calculator — Area of common two-dimensional shapes.
- Scientific Calculator — Evaluate expressions with functions, constants, powers and factorials.
- Pie Chart Calculator — Shares and slice angles for a set of values.
- Rational or Irrational Calculator — Whether the k-th root of a whole number is rational.
- Derivative Calculator — Derivative of a function of x, with a numeric check.
Sources
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.