Convert (x, y) to (r, θ)
Enter Cartesian coordinates below to instantly convert rectangular coordinates into polar form.
What this Cartesian to Polar Coordinates Calculator does
This calculator converts a point from Cartesian coordinates (x, y) to polar coordinates (r, θ). If you are working in geometry, trigonometry, physics, engineering, robotics, or computer graphics, this conversion is a common step. Instead of manually computing square roots and inverse tangents, you can type values and get a clean result instantly.
Cartesian vs polar coordinates
In a Cartesian system, a point is described by horizontal and vertical distance from the origin:
- x: horizontal displacement
- y: vertical displacement
In a polar system, the same point is described by:
- r: distance from origin
- θ (theta): angle from the positive x-axis
Formula used for conversion
The calculator uses the standard rectangular-to-polar formulas:
- r = √(x² + y²)
- θ = atan2(y, x)
The atan2 function is important because it handles all quadrants correctly and avoids ambiguity that can happen with a simple arctangent ratio.
Angle output options
- Degrees: θ in °
- Radians: θ in rad
- Normalized: forces angle into a positive range
How to use the calculator
- Enter the x-coordinate.
- Enter the y-coordinate.
- Choose degrees or radians.
- Select decimal precision.
- Optionally normalize the angle.
- Click Calculate.
Worked examples
Example 1: (3, 4)
r = √(3² + 4²) = √25 = 5
θ = atan2(4, 3) ≈ 53.1301°
So polar form is approximately (5, 53.1301°).
Example 2: (-2, 2)
r = √((-2)² + 2²) = √8 ≈ 2.8284
θ is in Quadrant II, so θ = 135° (or 3π/4 radians).
Polar form: (2.8284, 135°).
Example 3: (0, -7)
r = 7, and θ = -90° (or 270° if normalized).
Polar form can be written as (7, -90°) or (7, 270°).
Common mistakes to avoid
- Using
tan⁻¹(y/x)without checking the quadrant. - Mixing radians and degrees in the same calculation.
- Forgetting that the origin (0,0) has radius 0 and angle is convention-based.
- Rounding too early during multi-step calculations.
Why this matters in real applications
Coordinate conversion appears in vector analysis, signal processing, navigation, circular motion, and machine vision. Many systems measure direction and magnitude naturally in polar form, even when data is initially captured as x-y components.
Quick FAQ
Can the radius be negative?
Standard form uses r ≥ 0. A negative radius can represent the same point with a 180° angle shift, but this calculator returns non-negative radius.
Why do I get different but equivalent angles?
Angles are periodic. For example, 30°, 390°, and -330° all point in the same direction.
What happens at (0,0)?
Radius is 0. The angle is mathematically undefined, but many tools report 0 by convention.