Instant Cartesian (x, y) to Polar (r, θ) Conversion
Enter your rectangular coordinates below. This tool calculates radius r and angle θ using atan2(y, x) for correct quadrant handling.
What this Cartesian to Polar calculator does
This calculator converts standard Cartesian coordinates (x, y) into polar coordinates (r, θ). It is useful for algebra, trigonometry, calculus, physics, robotics, signal processing, and any context where points are easier to describe using a distance and direction.
Instead of manually squaring, square-rooting, and worrying about quadrants, you can input two numbers and get a clean, accurate answer instantly.
Cartesian vs polar coordinates
Cartesian (rectangular) coordinates
A point is written as (x, y), where:
- x is horizontal position (left/right),
- y is vertical position (down/up).
Polar coordinates
The same point is written as (r, θ), where:
- r is the distance from the origin,
- θ is the angle from the positive x-axis.
Polar form is especially convenient when motion is circular, rotational, or radial.
Formulas used
For a point (x, y), conversion formulas are:
- r = √(x² + y²)
- θ = atan2(y, x)
The atan2 function is important because it determines the correct quadrant automatically, unlike plain arctangent tan⁻¹(y/x), which can be ambiguous.
Step-by-step example
Example: (x, y) = (3, 4)
- Compute radius: r = √(3² + 4²) = √25 = 5
- Compute angle: θ = atan2(4, 3) ≈ 53.1301°
- Polar coordinates: (5, 53.1301°)
You can also express the angle in radians: θ ≈ 0.9273 rad.
Quadrant and axis notes
- Quadrant I: x > 0, y > 0
- Quadrant II: x < 0, y > 0
- Quadrant III: x < 0, y < 0
- Quadrant IV: x > 0, y < 0
- If x = 0 or y = 0, the point lies on an axis.
At the origin (0, 0), radius is zero and angle is mathematically undefined. Many calculators report angle as 0 by convention.
Common mistakes to avoid
- Using tan⁻¹(y/x) without fixing quadrant.
- Mixing degrees and radians in later calculations.
- Forgetting that multiple angles can represent the same direction (e.g., 45°, 405°, and -315°).
- Rounding too early and carrying that error into later steps.
Why this tool is useful
If you are solving homework, building engineering models, plotting vectors, or checking hand calculations, this coordinate converter provides quick and dependable output with consistent formatting.