Polar to Cartesian Converter

About the Polar to Cartesian Converter

Converting from polar to Cartesian coordinates transforms a point described by distance and angle (r, θ) into horizontal and vertical distances (x, y). The conversion uses the fundamental trigonometric relations x = r·cos(θ) and y = r·sin(θ).

This converter handles Polar → Cartesian, Cartesian → Polar, and batch conversions. It supports both degree and radian angle measures, visualizes the point on a coordinate plot, and provides a complete reference table. The batch mode efficiently processes multiple points for applications like plotting polar curves or converting data sets.

Polar-to-Cartesian conversion is essential in navigation (compass bearings to map positions), physics (force decomposition, orbital mechanics), signal processing (phasor analysis), computer graphics (rotation and translation), and robotics (sensor data processing). Understanding both coordinate systems and converting fluently between them is a core skill in mathematics and engineering. When you review a conversion, make sure the angle unit is correct and that the point lands in the expected quadrant.

When This Page Helps

Evaluating r·cos(θ) and r·sin(θ) at arbitrary angles — especially non-standard ones like 127° or 3.7 radians — requires a calculator anyway, and doing batch conversions by hand is impractical. This converter computes x and y from any (r, θ) input, correctly handles both degree and radian modes, converts multiple points in batch mode, and plots each result on a coordinate diagram. It is the fastest way to convert polar data to Cartesian for graphing, physics problems, or programming applications.

How to Use the Inputs

1. Choose conversion direction: Polar → Cartesian, Cartesian → Polar, or Batch.
2. Select angle units (degrees or radians).
3. Enter r and θ for polar input, or x and y for Cartesian input.
4. For batch mode, enter r,θ pairs separated by semicolons.
5. Use presets for standard angles like 45°, 90°, 180°.
6. Review the output cards for all converted values.
7. Check the coordinate plot for visual confirmation.

Example Calculation

Tips & Best Practices

• At 0° (or 360°): x = r, y = 0. At 90°: x = 0, y = r.
• Negative r reflects the point through the origin: (−r, θ) = (r, θ+180°).
• π radians = 180°. Remember: radians = degrees × π/180.
• Complex numbers in polar form: z = r·e^(iθ) converts to z = x + iy.
• For batch conversion, ensure consistent angle units (all degrees or all radians).

Polar Curves and Their Cartesian Equivalents

Many elegant curves have simple polar equations but complex Cartesian forms. The cardioid r = 1 + cos(θ) and the rose curve r = cos(nθ) are easy to describe in polar coordinates. Limaçons, spirals of Archimedes (r = aθ), and logarithmic spirals (r = ae^(bθ)) all have natural polar descriptions. Converting them to Cartesian form involves substituting x = r·cos(θ) and y = r·sin(θ), often yielding implicit equations. Understanding both representations gives deeper insight into curve geometry.

Applications in Navigation and Radar

Radar and sonar systems naturally output data in polar form — a distance (range) and angle (bearing) from the sensor. Converting to Cartesian coordinates places targets on a standard map grid. Aviation uses polar notation (heading and distance from a waypoint), and maritime navigation measures bearing and range to landmarks. Weather radar displays precipitation in polar coordinates centered on the antenna, then converts to geographic coordinates for map overlays.

Polar to Cartesian in Programming

Game development and computer graphics frequently convert between coordinate systems. A character "facing 45° and moving 10 units" is naturally described in polar (r=10, θ=45°), but screen rendering needs pixel coordinates (x=7.07, y=7.07). Physics engines decompose velocity vectors from speed + angle into vx and vy components. Most programming languages provide cos() and sin() functions that expect radians, so degree-to-radian conversion (θ_rad = θ_deg × π/180) is a common prerequisite step.

Frequently Asked Questions

• How do I convert polar to Cartesian?

  Use x = r·cos(θ) and y = r·sin(θ). Make sure θ is in the correct unit (degrees or radians).

• What if θ is in radians?

  The formulas are the same — just ensure your calculator/code is in radian mode. 1 radian ≈ 57.3°.

• Can r be negative?

  Technically yes — (−r, θ) represents the same point as (r, θ + π), reflected through the origin.

• What is the connection to complex numbers?

  The polar form of a complex number z = r(cos θ + i sin θ) = r·e^(iθ) directly gives x = r cos θ and y = r sin θ, so the same trig rules drive both coordinate conversion and complex-plane rotation.

• When is polar form preferred?

  For problems with rotational symmetry, angular motion, spirals, and oscillations. Also for complex multiplication and division.

• How do I handle angles beyond 360°?

  Reduce modulo 360° (or 2π radians). The point (r, θ+360°) is identical to (r, θ).