Polar to Rectangular Calculator

What Is a Polar to Rectangular Calculator?

A Polar to Rectangular Calculator converts a point or complex number from polar form (r, θ) to Cartesian (rectangular) coordinates (x, y). In electrical engineering and signal processing, the same conversion maps a phasor r∠θ to the complex number z = x + i y (or x + j y). The calculator handles both angle units—degrees and radians—normalizes negative angles and wrap‑around, and optionally rounds the results to your chosen precision. This saves time when plotting vectors, adding phasors, resolving forces, or evaluating complex impedances. Behind the scenes, the math is just trigonometry: multiply the magnitude by cosine for x and by sine for y. Clear, responsive formula rendering shows each step so students and professionals can audit results quickly.

About the Polar to Rectangular Calculator

The tool accepts r ≥ 0 and an angle θ in either degrees or radians. It converts degrees to radians when necessary, computes x and y, and formats both as decimals (and optionally as surds for special angles). Because periodicity means θ and θ + 2πk represent the same direction, the calculator also returns a normalized angle in (−π, π] or [0, 2π) for reference. If you’re working with complex numbers, it displays z = x + i y and the magnitude/angle check so you can verify that converting back recovers the original inputs. For vector work, the same equations resolve a vector of length r at heading θ into orthogonal components.

How to Use This Polar to Rectangular Calculator

1. Enter magnitude r and angle θ. Choose angle units (degrees or radians).
2. Pick output precision. Optionally enable normalized angle display for reference.
3. Submit to compute x, y, and the complex form z = x + i y. Copy results into your worksheet or code.
4. Use component form to add vectors or phasors by summing x and y separately, then (optionally) convert back to polar.

Examples

• Classic 3–4–5: r = 5, θ = 53.13° → x ≈ 3.00, y ≈ 4.00.
• Radians: r = 4, θ = π/6 → x ≈ 3.464, y = 2.
• Negative angle: r = 2, θ = −30° → x ≈ 1.732, y = −1.
• Third quadrant: r = 10, θ = 225° → x ≈ −7.071, y ≈ −7.071.

Formula Snippets Ready for Rendering

FAQs

What’s the difference between polar and rectangular coordinates?

Polar uses magnitude and angle (r, θ); rectangular uses orthogonal components (x, y). They represent the same point.

How do I choose degrees or radians?

Use degrees for everyday angles; radians are standard in calculus and many programming libraries. The calculator converts either way.

Can r be negative?

Conventionally r ≥ 0. A negative r can be rewritten with r' = |r| and θ' = θ + π.

What if my angle exceeds 360° or 2π?

Angles are periodic. Normalize by adding or subtracting 360° (or 2π) until the angle falls within your preferred range.

Does this work for complex numbers?

Yes. r∠θ converts to z = x + i y using the same equations; i (or j) denotes the imaginary unit.

Why do my results differ slightly from a calculator?

Small differences come from rounding and angle unit assumptions. Ensure you’ve selected the correct degrees/radians setting.

Can I add vectors in polar form?

Convert each to rectangular, add x’s and y’s separately, then convert the sum back to polar for magnitude and angle.

How precise should I make the output?

Use enough decimals for your task—engineering often uses 3–4 significant figures; graphics may require more.

What about special angles like 30°, 45°, 60°?

The calculator can display exact surds (√3/2, 1/√2) alongside decimals for educational clarity.

Is θ measured from the positive x‑axis?

Yes, standard polar coordinates measure θ counterclockwise from the positive x‑axis unless specified otherwise.

Can I input angles in grads or turns?

Convert grads/turns to radians or degrees first, then enter the converted value. Future versions may add direct support.