Polar to Rectangular Calculator
Convert polar coordinates (r, θ) to rectangular (x, y) and back — with step-by-step working, quadrant-correct atan2 handling, degree/radian toggle, and complex number polar form conversion.
Conversion Result
Convert complex numbers between rectangular form a + bi and polar form r·cis(θ) / r·eiθ. Same math as coordinate conversion — applied to the complex plane.
Click any row to load that angle into the converter instantly.
| θ (deg) | θ (rad) | r | x = r·cos(θ) | y = r·sin(θ) | Quadrant |
|---|
Polar vs Rectangular Coordinates — What's the Difference?
This polar to rectangular calculator converts between two fundamental coordinate systems used throughout mathematics, physics, and engineering. Understanding both systems — and when to use each one — is essential for calculus, complex analysis, and physics.
Rectangular coordinates (x, y) — also called Cartesian coordinates — locate a point by its horizontal distance x from the origin and vertical distance y. They are natural for algebraic equations and linear motion.
Polar coordinates (r, θ) locate a point by its radial distance r from the origin and the angle θ it makes with the positive x-axis. They are natural for circular motion, waves, and anything with rotational symmetry.
| Property | Rectangular (x, y) | Polar (r, θ) |
|---|---|---|
| Locates by | Horizontal + vertical distance | Distance + angle from origin |
| Natural for | Lines, polygons, linear equations | Circles, spirals, rotations |
| Unique representation | Yes | No (θ + 2π same point) |
| Circle equation | x² + y² = r² | r = constant (simple!) |
| Used in | Algebra, linear algebra | Calculus, physics, complex analysis |
How to Convert Polar to Rectangular — Step-by-Step
The polar to rectangular conversion uses basic trigonometry. For a point at distance r from the origin at angle θ, its horizontal component is r·cos(θ) and its vertical component is r·sin(θ).
Example 1 — Q1 Point: Polar (5, 53.13°) → Rectangular
- Given: r = 5, θ = 53.13°
- x = r·cos(θ) = 5·cos(53.13°) = 5 × 0.6 = 3
- y = r·sin(θ) = 5·sin(53.13°) = 5 × 0.8 = 4
- Rectangular: (3, 4) — in Q1 (both positive) ✓
- Verify: √(3² + 4²) = √25 = 5 = r ✓
Example 2 — Q2 Point: Polar (5, 126.87°) → Rectangular
- Given: r = 5, θ = 126.87°
- x = 5·cos(126.87°) = 5 × (−0.6) = −3
- y = 5·sin(126.87°) = 5 × 0.8 = 4
- Rectangular: (−3, 4) — in Q2 (x neg, y pos) ✓
Example 3 — Radians: Polar (2, π/3) → Rectangular
- Given: r = 2, θ = π/3 ≈ 1.0472 rad
- x = 2·cos(π/3) = 2 × 0.5 = 1
- y = 2·sin(π/3) = 2 × (√3/2) = √3 ≈ 1.7321
- Rectangular: (1, √3) ✓
How to Convert Rectangular to Polar — Step-by-Step
The rectangular to polar conversion uses the Pythagorean theorem for r and the atan2 function for θ. Using plain atan(y/x) instead of atan2(y, x) is one of the most common errors in coordinate conversion.
⚠️ atan2 vs atan — Critical Distinction: atan(y/x) only returns −90° to 90° (Q1 and Q4 only). For a Q2 point like (−3, 4): atan(4/−3) = atan(−1.333) ≈ −53.13° — WRONG! atan2(4, −3) ≈ 126.87° — CORRECT! Always use atan2 for quadrant-correct results.
Example 1 — Q1: Rectangular (3, 4) → Polar
- r = √(x² + y²) = √(9 + 16) = √25 = 5
- θ = atan2(4, 3) ≈ 53.13° (0.9273 rad)
- Polar: (5, 53.13°) ✓
Example 2 — Q2: Rectangular (−3, 4) → Polar
- r = √(9 + 16) = 5
- θ = atan2(4, −3) ≈ 126.87°
- Note: atan(4/−3) = −53.13° — wrong quadrant! atan2 gives correct 126.87°
- Polar: (5, 126.87°) ✓
Example 3 — Q3: Rectangular (−3, −4) → Polar
- r = √(9 + 16) = 5
- θ = atan2(−4, −3) ≈ −126.87° (or 233.13°)
- Polar: (5, −126.87°) ✓
Example 4 — Axes: Rectangular (0, 5) → Polar
- r = √(0 + 25) = 5
- θ = atan2(5, 0) = 90°
- Polar: (5, 90°) ✓
Polar Form of Complex Numbers
Every complex number a + bi corresponds to a point (a, b) in the complex plane. Its polar form expresses the same number using the modulus r (distance from origin) and argument θ (angle with positive real axis).
The three notations — r·cis(θ), r·e^(iθ), and r(cos θ + i·sin θ) — are all equivalent. The exponential form r·e^(iθ) is most compact and most useful for multiplication and powers of complex numbers.
Example: 3 + 4i in Polar Form
- a = 3, b = 4
- r = √(3² + 4²) = √25 = 5
- θ = atan2(4, 3) ≈ 53.13° (0.9273 rad)
- Polar form: 5·cis(53.13°) = 5·e^(i·0.9273)
Why polar form is powerful: Multiplying complex numbers: r₁·r₂ and θ₁+θ₂ (multiply moduli, add arguments). Powers: (r·cis(θ))ⁿ = rⁿ·cis(nθ). De Moivre's theorem. Roots: find all nth roots geometrically spaced around a circle.
Common Mistakes — Quadrant Errors and Degree/Radian Mix-ups
❌ Mistake 1 — Using atan instead of atan2
- Point (−3, 4) is in Q2 → correct θ ≈ 126.87°
- ❌ atan(4/−3) = atan(−1.333) ≈ −53.13° — wrong quadrant!
- ✅ atan2(4, −3) ≈ 126.87° — correct!
- Fix: Always use atan2(y, x) — pass x and y separately, not y/x as a single ratio.
❌ Mistake 2 — Degree/Radian Mix-up
- ❌ cos(90) in radians = cos(90 rad) ≈ −0.448 — treating degrees as radians
- ✅ cos(90°) = cos(π/2 rad) = 0 — correct
- Fix: Always check your calculator's mode. Convert degrees to radians: multiply by π/180.
❌ Mistake 3 — Negative r
- r is always ≥ 0. If your calculation gives r < 0, use |r| and add 180° to θ.
- ❌ Polar (−5, 30°) — r cannot be negative in standard form
- ✅ Equivalent standard form: (5, 210°) — same point!
❌ Mistake 4 — Forgetting Angle Periodicity
- The same point has infinitely many polar representations: (r, θ) = (r, θ+360°) = (r, θ+720°)…
- Standard convention: θ in (−180°, 180°] or [0°, 360°)
- This calculator uses (−180°, 180°] — the standard mathematical convention.
❌ Mistake 5 — Points on the Axes
- On positive x-axis: (r, 0) = (r, 0°) — cos is 1, sin is 0
- On positive y-axis: (r, 90°) — cos is 0, sin is 1
- At origin: r = 0, θ is undefined (any angle works)
- atan2(0, 0) is technically undefined — this calculator handles it gracefully.
Worked Examples — All Four Quadrants
| Rectangular (x, y) | r = √(x²+y²) | θ = atan2(y,x) | Quadrant |
|---|---|---|---|
| (3, 4) | 5 | 53.13° | Q1 |
| (−3, 4) | 5 | 126.87° | Q2 |
| (−3, −4) | 5 | −126.87° | Q3 |
| (3, −4) | 5 | −53.13° | Q4 |
| (1, 0) | 1 | 0° | +x axis |
| (0, 1) | 1 | 90° | +y axis |
| (−1, 0) | 1 | 180° | −x axis |
| (0, −1) | 1 | −90° | −y axis |
| (1, 1) | √2 | 45° | Q1 |
| (5, 0) | 5 | 0° | +x axis |
Frequently Asked Questions
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| θ (deg) | cos θ | sin θ |
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