Polar coordinates calculator: convert (x, y) to (r, θ) and back
This polar coordinates calculator uses the relations from OpenStax Precalculus: x = r cos θ, y = r sin θ and r² = x² + y². Example: the point (3, 3) has r = 3√2 ≈ 4.2426 and θ = 45° (π/4), and the polar point (3, 90°) is (0, 3).
Example: Convert: rectangular (x, y) → polar (r, θ) · x 3 · y 3 · Angle unit: degrees → Result: r = 4.2426, θ = 45°. Source: OpenStax Precalculus 2e, 8.3 Polar Coordinates: x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x; Examples (3, π/2) → (0, 3), (−2, 0) → (−2, 0), (3, 3) → (3√2, π/4) (read 09.10.2026). Updated: .
How it is calculated
x = r cos θ, y = r sin θ – and back: r = √(x² + y²), tan θ = y/x. A polar point gives the distance r from the origin (the pole) and the angle θ from the positive x-axis. Converting to polar needs care with the quadrant: tan θ = y/x alone cannot tell (3, 3) from (−3, −3). The calculator uses the signs of x and y and returns θ between 0° and 360°.
Examples
| Given | Result |
|---|---|
| (x, y) = (3, 3) | r = 4.2426 (3√2), θ = 45° = 0.7854 rad (π/4) |
| (x, y) = (−4, 3) | r = 5, θ = 143.1301° = 2.4981 rad |
| (x, y) = (1, −1) | r = 1.4142, θ = 315° |
| (x, y) = (0, −2) | r = 2, θ = 270° |
| (r, θ) = (3, 90°) | (x, y) = (0, 3) |
| (r, θ) = (−2, 0°) | (x, y) = (−2, 0) |
| (r, θ) = (−1, 120°) | (x, y) = (0.5, −0.866) |
| (r, θ) = (5, 30°) | (x, y) = (4.3301, 2.5) |
The first, fifth and sixth rows are OpenStax examples 3, 4 and 5; (−1, 2π/3) is one of its exercises, with the answer (1/2, −√3/2).
One point, many polar forms
OpenStax points out that (3√2, π/4), (−3√2, 5π/4) and (3√2, −7π/4) are the same point. Adding 360° to θ, or making r negative and turning by 180°, does not move it. The values of sine and cosine for common angles are on the unit circle calculator; complex numbers in polar form are handled by the complex number calculator.
Frequently asked questions
How do I convert rectangular to polar coordinates?
r = √(x² + y²) and θ from tan θ = y/x in the right quadrant. (3, 3): r = √18 = 4.2426, θ = 45°.
How do I convert polar to rectangular coordinates?
x = r cos θ and y = r sin θ. (3, 90°) gives x = 0 and y = 3.
Why is the angle for (−4, 3) not −36.87°?
Because the point lies in the second quadrant. tan⁻¹(3/−4) gives −36.87°, and adding 180° gives the correct 143.1301°.
Can r be negative?
Yes, a negative r points the opposite way: (−2, 0°) is the point (−2, 0). The calculator always returns r ≥ 0 when converting to polar.
Degrees or radians?
Both. 45° = π/4 ≈ 0.7854 rad; switch the angle unit to radians for input and output.
How exact are the results?
They are calculated values rounded to four decimals, so √2 appears as 1.4142.
Sources and legal basis
As of: · Editorial policy
Related tools
- Unit Circle CalculatorUnit circle calculator: for any angle in degrees or radians it gives sine, cosine, tangent, the point on the circle, the quadrant and the exact radical values.
- Complex Number CalculatorComplex number calculator: add, subtract, multiply, divide, raise to a power and take roots, with polar form, Euler form, modulus and the steps shown.
- Sin Cos Tan CalculatorSin cos tan calculator: enter an angle in degrees, radians or gradians and see all trig functions – or find the angle from a value with arcsin, arccos, arctan.
- Pythagorean Theorem CalculatorPythagorean theorem calculator: enter two sides to get the third side, both angles, area and height – with exact square roots, working and a right-angle check.
- Vector Calculator with StepsVector calculator for 2D and 3D: sum, dot product, cross product, magnitude, angle and projection of two vectors – instant, with the working shown step by step.
- Online CalculatorCalculator online, free and instant: type or tap and see the answer as you go – order of operations, brackets, percent, square root, steps and a history list.
- Grade curve calculatorGrade curve calculator for teachers: paste the scores, pick a curve – raise the mean, scale to the top score, square root or bell – and see each new score.
- Attendance percentage calculatorAttendance percentage calculator for school and college: enter classes held and attended, and see how many you can miss or must attend to reach 75%.
- GWA Calculator (General Weighted Average)GWA calculator for college in the Philippines: enter each subject’s grade and units and get your general weighted average with the full computation.