Quick answer
To convert polar (r, θ) to rectangular coordinates, x = r cos θ and y = r sin θ. For r = 5 and θ = 53.13° the point is (3, 4).
Enter the radius and the angle. The calculator returns x and y.
Related tools: Cartesian to Polar Converter: Radius and angle from x and y. Rectangular Prism Volume Calculator: Volume of a rectangular prism (box). Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results.
How to use the Polar to Rectangular Converter
- Enter radius r.
- Enter angle θ (°).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Radius r | A number | — |
| Angle θ | A number, in ° | — |
Formula and method
Formula
x = r cos θ; y = r sin θ
IEEE-754 double precision; shown to the decimals indicated.
Angle in degrees.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Radius r = 5; Angle θ = 53.13010235.
| Result | Value |
|---|---|
| x | 3 |
| y | 4 |
Example 2
Inputs: Radius r = 2; Angle θ = 180.
| Result | Value |
|---|---|
| x | -2 |
| y | 0 |
Example 3
Inputs: Radius r = 10; Angle θ = -45.
| Result | Value |
|---|---|
| x | 7.071068 |
| y | -7.071068 |
The first line is the headline result (x). Below it the polar to Rectangular Converter lists y, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Angle units
Use degrees unless you have radians; the angle 53.13° is 0.9273 rad. The same r with θ = 126.87° gives x = −3 and y = 4, in the second quadrant.
Negative radius
A negative radius flips the point to the opposite side of the origin; (−5, 53.13°) is (−3, −4).
Quadrants and the angle unit
With r = 5 and an angle of about 53.13° the point is (3, 4). Angles beyond 90° move into other quadrants: r = 2 and θ = 135° gives x ≈ −1.4142 and y ≈ 1.4142.
Enter the angle in the unit the calculator expects (degrees, as the example shows). Mixing degrees and radians is the most common cause of wrong coordinates.
How the result changes with radius r
Holding every other input at the first worked example, this table changes only radius r. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Radius r | x | y |
|---|---|---|
| 2.5 | 1.5 | 2 |
| 5 | 3 | 4 |
| 10 | 6 | 8 |
| 20 | 12 | 16 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Radius r: left empty shows "Enter radius r." beside that field and no result.
- Angle θ: left empty shows "Enter angle θ." beside that field and no result.
Choosing between the Polar to Rectangular 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 |
|---|---|---|
| [Cartesian to Polar Converter](/math/cartesian-to-polar/) | x and y | Radius r |
| [Rectangular Prism Volume Calculator](/math/rectangular-prism-volume/) | length, width and height | Volume |
| [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 |
Assumptions and limitations
- Negative radii point the opposite way
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 radius r and angle θ to confirm this.
Frequently asked questions
What if the angle exceeds 360°?
It wraps around: 413.13° gives the same point as 53.13°.
Is the result exact?
Rounded to a number of decimal places.
How to go the other way?
Use the Cartesian to Polar calculator.
How do I convert back to polar form?
Use r = √(x² + y²) and θ = atan2(y, x), which keeps the correct quadrant.
What are the coordinates for r = 1 and θ = 90°?
The point is (0, 1). The x value is 0 up to rounding.
What does the Polar to Rectangular Converter calculate?
x and y from radius and angle. It returns x and y.
What formula does the Polar to Rectangular Converter use?
x = r cos θ; y = r sin θ. Angle in degrees.
What do I need to enter in the polar to Rectangular Converter?
You enter Radius r and Angle θ (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Negative radii point the opposite way.
What is an example of a polar to Rectangular result?
With radius r 5 and angle θ 53.13010235, the x is 3.
Related calculators
- Cartesian to Polar Converter — Radius and angle from x and y.
- Rectangular Prism Volume Calculator — Volume of a rectangular prism (box).
- 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.
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.