Polar to Rectangular Coordinates Converter

About the Polar to Rectangular Coordinates Converter

The **Polar to Rectangular Coordinates Converter** performs bidirectional conversion between polar coordinates (r, θ) and rectangular (Cartesian) coordinates (x, y). Toggle between "Polar → Rectangular" and "Rectangular → Polar" modes to convert in either direction, with the angle displayed in degrees, radians, and gradians simultaneously.

Polar coordinates express a point as a distance from the origin (r) and an angle from the positive x-axis (θ). Rectangular coordinates express the same point as horizontal (x) and vertical (y) displacements. The conversion formulas are: x = r·cos(θ) and y = r·sin(θ) for polar-to-rectangular, and r = √(x² + y²) and θ = atan2(y, x) for rectangular-to-polar.

It gives far more than a simple conversion. An SVG coordinate visualization plots the point, draws the radius line from the origin, shows dashed projection lines to the axes, and marks the angle arc — all updating in real time. A quadrant indicator grid highlights which quadrant the point falls in, with sign rules displayed. The unit-circle reference table lists x and y coordinates for 14 standard angles on the unit circle, highlighting the row matching your current angle. A batch-conversion table shows the result of your current angle applied to eight common radii.

Eight preset buttons cover the most common conversions in both directions, including negative coordinates and multi-quadrant examples. Angle units can be set to degrees or radians, and the decimal precision slider controls all output formatting.

When This Page Helps

Polar to Rectangular Coordinates Converter helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like x, y, r (radius) in one pass.

How to Use the Inputs

1. Enter the required inputs (Conversion Mode, Angle Unit, Decimal Precision).
2. Complete the remaining fields such as r (radius), x, y.
3. Review the output cards, especially x, y, r (radius), θ (degrees).
4. Compare the result with the formula, diagram, or example values to catch sign, unit, or rounding mistakes.

Example Calculation

Tips & Best Practices

• atan2(y, x) correctly handles all four quadrants, unlike atan(y/x) which only covers Q1 and Q4.
• A negative r in polar coordinates reflects the point through the origin.
• The origin (0, 0) in rectangular is (0, undefined) in polar — θ is undefined when r = 0.
• Gradians divide a full rotation into 400 units instead of 360° or 2π.
• Complex numbers in polar form use the same conversion: z = r·(cos θ + i·sin θ).

What This Polar to Rectangular Coordinates Converter Solves

This calculator is tailored to polar to rectangular coordinates converter workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.

How To Interpret The Outputs

Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.

Study And Practice Strategy

A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.

Frequently Asked Questions

• What are polar coordinates?

  Polar coordinates (r, θ) specify a point by its distance from the origin (r) and its angle from the positive x-axis (θ), measured counterclockwise.

• How do I convert polar to rectangular?

  Use x = r·cos(θ) and y = r·sin(θ). Ensure your angle is in the correct unit (degrees or radians) before applying the formulas.

• How do I convert rectangular to polar?

  Use r = √(x² + y²) for the radius and θ = atan2(y, x) for the angle. atan2 returns the correct angle in all quadrants.

• What is the difference between atan and atan2?

  atan(y/x) only returns angles in (−90°, 90°), losing quadrant information. atan2(y, x) uses both signs to return the correct angle in (−180°, 180°].

• Can r be negative?

  Yes. A negative r means the point is reflected through the origin. (−r, θ) is the same as (r, θ + 180°). This calculator handles both positive and negative r.

• What is the connection to complex numbers?

  A complex number z = x + iy can be written in polar form as z = r·(cos θ + i·sin θ) = r·e^(iθ). The conversion formulas are identical.