Convert (x, y) to (r, θ)
Enter Cartesian coordinates and instantly convert them to polar form.
What this Cartesian to Polar calculator does
This tool converts a point from Cartesian coordinates (x, y) into polar coordinates (r, θ). In Cartesian form, a point is measured by horizontal and vertical distance. In polar form, the same point is measured by distance from the origin and angle from the positive x-axis.
If you are working with vectors, trigonometry, complex numbers, physics, engineering, robotics, or signal processing,
this conversion comes up constantly. The calculator uses atan2(y, x) so the angle is placed in the correct quadrant automatically.
Formulas used
Radius: r = √(x² + y²)
Angle (radians): θ = atan2(y, x)
Angle (degrees): θ° = θ × (180 / π)
The key detail is atan2, not just arctan(y/x). The regular arctangent can lose quadrant information. atan2 avoids that issue and returns the principal angle correctly based on signs of both x and y.
How to convert Cartesian to polar manually
Step 1: Compute the radius
Square x and y, add them, and take the square root. This gives the distance from the origin: r = √(x² + y²).
Step 2: Compute the angle
Use θ = atan2(y, x). This gives an angle typically in the range (−π, π] in radians. If needed, convert to degrees by multiplying by 180/π.
Step 3: State equivalent angle(s)
Polar angles are periodic. You can add or subtract full rotations and represent the same point:
- In degrees: θ, θ + 360°, θ − 360°, ...
- In radians: θ, θ + 2π, θ − 2π, ...
Examples
Example 1: (3, 4)
r = √(3² + 4²) = 5. θ = atan2(4, 3) ≈ 53.13° (or 0.9273 rad). So polar form is approximately (5, 53.13°).
Example 2: (-2, 2)
r = √((-2)² + 2²) = √8 ≈ 2.8284. θ = atan2(2, -2) = 135° (or 2.3562 rad), which is in Quadrant II.
Example 3: (0, -7)
r = 7. θ = -90° (or -π/2 rad). The point lies on the negative y-axis.
Important edge case: the origin
For (0, 0), the radius is 0, but the angle is mathematically undefined because every angle points to the same location. Many systems display θ = 0 by convention. This calculator clearly flags that special case.
Common mistakes to avoid
- Using arctan(y/x) and getting the wrong quadrant.
- Forgetting whether your system expects degrees or radians.
- Rounding too early in multi-step calculations.
- Ignoring the fact that angles have infinitely many equivalent values.
Quick FAQ
Is polar form unique?
No. The radius is usually taken as nonnegative for a standard form, but the angle can differ by full rotations.
Can the angle be negative?
Yes. Negative angles are valid and represent clockwise rotation from the positive x-axis.
Why does this calculator provide an equivalent angle?
It helps verify periodicity and makes it easier to match textbook or software conventions.
Final note
Use this calculator to save time, reduce sign errors, and verify your hand calculations. Enter x and y, choose your preferred angle unit, and get a reliable polar conversion instantly.