Convert polar (r, θ) coordinates to Cartesian (x, y) and back, with batch mode, coordinate plot, and complete conversion reference.
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. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
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 instantly 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.
x = r · cos(θ) y = r · sin(θ) r = √(x² + y²) θ = atan2(y, x)
Result: x ≈ 3, y ≈ 4
x = 5·cos(53.13°) ≈ 3, y = 5·sin(53.13°) ≈ 4. This is the 3-4-5 triangle in polar form.
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.
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.
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.
Use x = r·cos(θ) and y = r·sin(θ). Make sure θ is in the correct unit (degrees or radians).
The formulas are the same — just ensure your calculator/code is in radian mode. 1 radian ≈ 57.3°.
Technically yes — (−r, θ) represents the same point as (r, θ + π), reflected through the origin. Use this as a practical reminder before finalizing the result.
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. Keep this note short and outcome-focused for reuse.
For problems with rotational symmetry, angular motion, spirals, and oscillations. Also for complex multiplication and division.
Reduce modulo 360° (or 2π radians). The point (r, θ+360°) is identical to (r, θ).