How rectangular to polar conversion works
A rectangular (Cartesian) coordinate describes a point by its horizontal offset x and vertical offset y from the origin (0, 0). A polar coordinate describes the same point by its distance r from the origin and its angle θ, measured counterclockwise from the positive x-axis.
Rectangular to polar formulas
- Radius: r = √(x² + y²)
- Angle: θ = atan2(y, x)
- Radians to degrees: deg = rad × (180 / π)
- Phasor form: r ∠ θ
Polar to rectangular formulas
- Horizontal component: x = r · cos(θ)
- Vertical component: y = r · sin(θ)
- Complex number: z = x + jy = r(cos θ + j sin θ)
- Euler form: z = r · e^(jθ)
Why the quadrant matters: atan2(y, x)
The single-argument arctan(y/x) only returns angles between −90° and 90°, so it cannot tell opposite quadrants apart: (−3, −4) gives (−4)/(−3) = 4/3, exactly the same as (3, 4). This calculator uses the two-argument function atan2(y, x), which checks the signs of x and y:
| Quadrant | Signs | Range of θ | Rule for θ |
|---|---|---|---|
| Quadrant I | x > 0, y > 0 | 0° – 90° | θ = arctan(y / x) |
| Quadrant II | x < 0, y ≥ 0 | 90° – 180° | θ = arctan(y / x) + 180° |
| Quadrant III | x < 0, y < 0 | 180° – 270° | θ = arctan(y / x) + 180° |
| Quadrant IV | x > 0, y < 0 | 270° – 360° | θ = arctan(y / x) + 360° |
Why use this calculator?
Built for students, teachers and electrical engineers who need fast, correct coordinate conversions:
- Free, nothing to install: works in any browser on desktop, tablet or phone.
- Step-by-step working: shows every intermediate value instead of a single number.
- Live graph: draws the point, the radius vector and the angle as you type.
- Both directions: switch between rectangular → polar and polar → rectangular with one click.