Cartesian to Polar Converter

About the Cartesian to Polar Converter

Converting between Cartesian (rectangular) and polar coordinate systems is a fundamental operation in mathematics, physics, and engineering. While Cartesian coordinates specify a point using horizontal (x) and vertical (y) distances from the origin, polar coordinates use a radius (r) — the distance from the origin — and an angle (θ) measured from the positive x-axis.

This converter handles both directions: Cartesian → Polar using r = √(x² + y²) and θ = atan2(y, x), and Polar → Cartesian using x = r·cos(θ) and y = r·sin(θ). The atan2 function correctly handles all four quadrants, unlike the basic arctan. A batch mode converts multiple points simultaneously, and a visual coordinate plot shows the point's location.

Polar coordinates are especially useful for problems with circular symmetry: orbits, spirals, wave patterns, antenna radiation patterns, and complex number operations. Understanding both systems and converting fluently between them is essential in calculus (double integrals), physics (orbital mechanics), signal processing, and computer graphics (rotation transformations).

When This Page Helps

Converting coordinates by hand requires evaluating square roots and inverse tangent functions while correctly identifying the quadrant — the basic arctan function only covers half the plane, making atan2 essential. Batch conversions of multiple points compound these issues. This converter handles all four quadrants correctly using atan2, supports both degree and radian modes, converts multiple points at once in batch mode, and plots each result visually. It is ideal for students checking trigonometry homework, engineers decomposing force vectors, and anyone working with polar curves or complex number arithmetic.

How to Use the Inputs

1. Choose the conversion direction: Cartesian → Polar, Polar → Cartesian, or Batch.
2. Select angle units — degrees or radians.
3. Enter coordinates (x, y) for Cartesian input or (r, θ) for polar input.
4. For batch mode, enter multiple x,y pairs separated by semicolons.
5. Use presets for common points like (3, 4) or (−1, √3).
6. Review the output cards for all converted values and quadrant identification.
7. Check the coordinate plot for a visual representation of the point.

Example Calculation

Tips & Best Practices

• Use atan2(y, x) instead of atan(y/x) to get the correct quadrant automatically.
• Polar angles are typically measured counterclockwise from the positive x-axis.
• To normalize angles, add 360° to negative results.
• In batch mode, the semicolon separates points while the comma separates x and y.
• For complex numbers, polar form is z = r·e^(iθ) = r(cos θ + i sin θ).

Why Two Coordinate Systems?

Cartesian coordinates excel at describing rectangular geometry — grids, boxes, linear motion. Polar coordinates excel at circular geometry — orbits, spirals, rotations. The choice depends on the problem's symmetry. Integrating over a circular region is painful in Cartesian but natural in polar (r dr dθ). Conversely, describing a rectangle is trivial in Cartesian but awkward in polar. Fluency in both systems and quick conversion between them is a core mathematical skill.

The atan2 Function and Quadrant Handling

The standard arctan(y/x) function cannot distinguish between quadrants I and III (both give positive y/x) or quadrants II and IV (both give negative y/x). The atan2(y, x) function solves this by examining the signs of both x and y independently, returning angles in the full range (−180°, 180°]. This is why every programming language and scientific calculator provides atan2 — it is the correct function for coordinate conversion.

Coordinate Conversion in Complex Numbers

Every complex number z = x + iy can be written in polar form z = r·e^(iθ), where r = |z| = √(x²+y²) and θ = arg(z) = atan2(y,x). Multiplication in polar form is elegant: z₁·z₂ = r₁r₂·e^(i(θ₁+θ₂)) — multiply magnitudes, add angles. This is why converting to polar form simplifies complex arithmetic, roots of unity calculations, and Fourier analysis. De Moivre's theorem [r·e^(iθ)]ⁿ = rⁿ·e^(inθ) is another powerful result that relies on polar representation.

Frequently Asked Questions

• What is the difference between Cartesian and polar coordinates?

  Cartesian uses (x, y) perpendicular distances; polar uses (r, θ) distance and angle from the origin.

• Why use atan2 instead of atan?

  atan2(y, x) returns the correct angle in all four quadrants (−180° to 180°), while atan(y/x) only covers −90° to 90°.

• Can r be negative in polar coordinates?

  Conventionally r ≥ 0, but some contexts allow negative r, meaning the point is in the opposite direction: (−r, θ) = (r, θ + 180°).

• How do I convert radians to degrees?

  Multiply by 180/π. Conversely, degrees to radians: multiply by π/180.

• When is polar form more useful?

  For problems with circular symmetry — orbits, spirals, rotation, complex multiplication, and polar integrals.

• What about 3D coordinates?

  In 3D, spherical coordinates (r, θ, φ) or cylindrical coordinates (r, θ, z) extend the polar concept.