🧭 Rectangular to Polar Converter – Complex Numbers & Cartesian Coordinates
The rectangular to polar converter transforms a point or complex number expressed in Cartesian form (x, y) into its equivalent polar representation r ∠ θ. Whether you are working with complex numbers, phasors in electrical engineering, or Cartesian coordinates in geometry, this tool instantly computes the modulus (magnitude) and argument (angle) — correctly handling all four quadrants and axis-aligned edge cases.
📐 Core Formulas
Two native math functions drive every conversion:
- Magnitude:
r = √(x² + y²)— computed asMath.hypot(x, y)for numerical stability. - Angle:
θ = atan2(y, x)— the quadrant-aware inverse tangent, returning values in(−π, π]radians.
Using atan2 instead of plain arctan(y/x) is critical: the regular arctan function only covers −90° to 90°, so a point in Quadrant III like (−3, −3) would be misidentified. atan2 always returns the correct full-circle angle.
📊 Output Forms Explained
| Form | Example (x=3, y=4) | Use Case |
|---|---|---|
| Polar | 5 ∠ 53.130° | Compact notation; phasors, navigation |
| Trigonometric | 5(cos 53.130° + i·sin 53.130°) | Complex number multiplication proofs |
| Exponential (Euler) | 5·e^(i·0.9273) | Signal processing, Fourier analysis |
🔄 Angle Normalization
The same direction can be expressed with different angle values. The tool supports two common conventions:
- Principal value
(-180°, 180°]or(−π, π]— the standard mathematical convention. Negative angles represent directions below the positive x-axis. - Positive range
[0°, 360°)or[0, 2π)— common in navigation, robotics, and engineering contexts where only non-negative angles are desired.
For example, (−1, −1) gives −135° in principal-value mode or 225° in positive-angle mode — both describe the same direction (Quadrant III).
🌟 Exact Angle Recognition
When the input corresponds to a common special angle — multiples of 30°, 45°, or 90° — the tool optionally displays the exact symbolic label (π/4, π/3, 2π/3, etc.) instead of a decimal approximation. This is especially helpful for students checking textbook answers or for signal-processing work where exact Euler-form notation is expected.
📋 Batch Conversion Mode
The batch mode tab lets you convert many coordinate pairs in one go. Enter one pair per line in any of these formats:
3, 4— comma-separated(−1, 1)— with parentheses3+4i— complex-number notation
Each valid row produces its own magnitude, angle, polar form, and quadrant classification. Results can be downloaded as a CSV file for use in spreadsheets or reports.
⚠️ Edge Cases & Special Values
- Origin (0, 0): Magnitude is
0; the angle is undefined (no direction exists for a zero vector). - Positive x-axis (y = 0, x > 0):
θ = 0° - Negative x-axis (y = 0, x < 0):
θ = ±180° - Positive y-axis (x = 0, y > 0):
θ = 90° - Negative y-axis (x = 0, y < 0):
θ = −90°
🔧 Common Applications
- Electrical engineering: Converting AC circuit phasors from rectangular
(R + jX)to polar impedance form|Z|∠φ. - Signal processing: Expressing complex Fourier coefficients in magnitude-and-phase notation.
- Physics: Converting velocity or force vectors from component form to magnitude-and-direction.
- Mathematics: Multiplying complex numbers using
r₁r₂ ∠ (θ₁ + θ₂)— far simpler in polar form than rectangular. - Navigation & robotics: Converting (Δx, Δy) displacement into range and bearing.
🎓 Relationship to the Inverse Conversion
This tool is the counterpart of the Polar to Rectangular converter. If you start with r and θ and need x and y, use the polar-to-rectangular tool instead. Together they provide a complete round-trip between the two representations, which is essential for understanding complex-plane geometry and for verifying calculations.