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