Polar to Cartesian Conversion
Convert between polar coordinates (r, theta) and Cartesian coordinates (x, y).
Includes formulas for both directions with examples.
Polar to Cartesian
x = r × cos(θ)
y = r × sin(θ)
y = r × sin(θ)
Polar coordinates use a distance from the origin (r) and an angle (θ) to locate a point. Cartesian coordinates use horizontal (x) and vertical (y) distances.
Cartesian to Polar
r = √(x² + y²)
θ = atan2(y, x)
θ = atan2(y, x)
Variables
| Symbol | Meaning |
|---|---|
| r | Radius — distance from the origin |
| θ | Angle measured from the positive x-axis (in radians or degrees) |
| x | Horizontal coordinate |
| y | Vertical coordinate |
Example 1 — Polar to Cartesian
Convert the polar point (5, 53.13°) to Cartesian coordinates.
x = r × cos(θ) = 5 × cos(53.13°) = 5 × 0.6 = 3
y = r × sin(θ) = 5 × sin(53.13°) = 5 × 0.8 = 4
(x, y) = (3, 4)
Example 2 — Cartesian to Polar
Convert the Cartesian point (-3, 4) to polar coordinates.
r = √((-3)² + 4²) = √(9 + 16) = √25 = 5
θ = atan2(4, -3) ≈ 126.87° (in the second quadrant)
(r, θ) = (5, 126.87°)
When to Use It
Use polar-Cartesian conversion for coordinate geometry:
- Converting between GPS-style (distance and bearing) and grid coordinates
- Working with circular motion, spirals, and curves in polar form
- Computer graphics and game development (rotation calculations)
- Plotting complex numbers on the complex plane
Key Notes
- Conversion formulas: Polar (r, θ) to Cartesian (x, y): x = r cosθ, y = r sinθ. Cartesian to polar: r = √(x² + y²), θ = atan2(y, x). The radius r is always non-negative.
- Use atan2, not plain arctan: The standard arctan function only returns angles in (−90°, 90°). The atan2(y, x) function returns angles in the full (−180°, 180°] range, correctly handling all four quadrants.
- Multiple polar representations: The same Cartesian point can be represented by infinitely many polar coordinates: (r, θ), (r, θ + 2π), (r, θ − 2π), etc. For negative r, (−r, θ + π) gives the same point.
- Useful for circles and spirals: Many curves are simpler in polar form. A circle of radius a centered at the origin is simply r = a. An Archimedean spiral is r = aθ — far simpler than the Cartesian equivalent.
- 3D extension — spherical coordinates: In 3D, polar coordinates extend to spherical (r, θ, φ) with x = r sinφ cosθ, y = r sinφ sinθ, z = r cosφ. These are natural for describing spheres, planets, and electromagnetic radiation patterns.