Convert between polar (r, θ) and rectangular (x, y) coordinates. Bidirectional conversion with quadrant visualization, unit circle reference, and batch table.
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.
This calculator provides 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.
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.
Polar → Rectangular: x = r·cos(θ), y = r·sin(θ). Rectangular → Polar: r = √(x²+y²), θ = atan2(y, x). The angle θ is measured counterclockwise from the positive x-axis.
Result: x ≈ 4.3301, y = 2.5
Using r=5, θ=30°, the calculator returns x ≈ 4.3301, y = 2.5. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
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.
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.
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.
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. Use this as a practical reminder before finalizing the result.
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.
Use r = √(x² + y²) for the radius and θ = atan2(y, x) for the angle. atan2 returns the correct angle in all quadrants.
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°].
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.
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.