Understanding Cartesian to Polar Conversion

Cartesian coordinates (x, y) and polar coordinates (r, θ) are two different systems for representing points in a 2D plane. While Cartesian coordinates use perpendicular axes, polar coordinates use distance from origin and angle from the positive x-axis.

Conversion Formulas

Radius (r): r = √(x² + y²)
Angle (θ): θ = atan2(y, x) [in radians]
θ (degrees) = θ (radians) × 180/π

Example Calculation

For point (3, 4):
r = √(3² + 4²) = √(9 + 16) = √25 = 5
θ = atan2(4, 3) ≈ 0.9273 radians ≈ 53.13°

Applications

Polar coordinates are particularly useful in fields involving circular or rotational systems, including physics (wave equations), engineering (control systems), and computer graphics (particle systems).

Frequently Asked Questions

What is the range of the angle θ?

The angle θ typically ranges from -π to π radians (-180° to 180°) when using the atan2 function, which determines the correct quadrant.

How are negative coordinates handled?

Negative x or y values are correctly processed by the atan2 function, which accounts for all four quadrants of the coordinate system.

What happens when both x and y are zero?

When both coordinates are zero, the radius r becomes 0, and the angle θ is undefined (often set to 0 by convention).

Cartesian to Polar Converter

Input Coordinates

Polar Coordinates

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Cartesian to Polar Converter - x y to r Theta

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