By convention r ≥ 0 (magnitude). Some textbooks allow r < 0 by reflecting across the origin (equivalent to adding π to θ); this calculator returns r ≥ 0 always.

What's the angle convention?

Standard mathematical convention: θ measured counter-clockwise from the positive x-axis. 0° = +x, 90° = +y, 180° = −x, 270° = −y. Engineering compass-bearing convention (0° = +y, clockwise) is DIFFERENT — convert if needed.

IEEE 754 atan2 specification; Spiegel MR (1968) "Mathematical Handbook of Formulas and Tables" Ch.1; standard coordinate-geometry textbooks.

By convention r ≥ 0 (magnitude). Some textbooks allow r < 0 by reflecting across the origin (equivalent to adding π to θ); this calculator returns r ≥ 0 always.

What's the angle convention?

Standard mathematical convention: θ measured counter-clockwise from the positive x-axis. 0° = +x, 90° = +y, 180° = −x, 270° = −y. Engineering compass-bearing convention (0° = +y, clockwise) is DIFFERENT — convert if needed.

IEEE 754 atan2 specification; Spiegel MR (1968) "Mathematical Handbook of Formulas and Tables" Ch.1; standard coordinate-geometry textbooks.

Polar ↔ Cartesian Coordinate Converter

Convert (x, y) ↔ (r, θ). Polar: r = √(x²+y²), θ = atan2(y, x). Cartesian: x = r·cos θ, y = r·sin θ.

Result

Polar (r, θ)

(5.0000, 53.1301°)

Quadrant I (++).

x3.000000
y4.000000
r (magnitude)5.000000
θ (degrees)53.130102°
θ (radians)0.927295
QuadrantI (++)
Reference angle (acute equiv)53.1301°

Step-by-step

r = √(3² + 4²) = √25 = 5.000000.
θ = atan2(4, 3) = 53.1301° = 0.927295 rad.

How to use this calculator

Pick direction. Cartesian-to-polar takes (x, y) and returns (r, θ).
Polar-to-Cartesian takes (r, θ in degrees) and returns (x, y).
The angle-unit selector switches the displayed θ between degrees and radians.

About this calculator

Two coordinate systems for the same plane. Cartesian (x, y) — perpendicular axes, intuitive for rectangular shapes. Polar (r, θ) — radius from origin + angle from positive x-axis, intuitive for circular shapes and rotation. Conversions: r = √(x²+y²); θ = atan2(y, x) — uses the TWO-argument arctangent so the quadrant is preserved (regular atan(y/x) loses the sign distinction between quadrants II/IV and I/III). x = r·cos θ; y = r·sin θ. Polar is the natural form for orbits, spirals, antenna patterns, and complex-number magnitude/phase representation; Cartesian is the natural form for translation, regular grids, and most CAD work.

Frequently asked

atan(y/x) returns an angle in (−π/2, π/2) — it can't distinguish Quadrant I from Quadrant III (both have positive y/x ratio) or II from IV. atan2(y, x) is a two-argument function that returns the angle in (−π, π] with the correct quadrant. Standard in every numerical library.