Converting…
Select a direction, enter your coordinates, and press Convert to see the result with a live plot.
What Are Polar and Rectangular Coordinates?
Every point in a 2D plane can be described in two fundamentally different ways. Rectangular (Cartesian) coordinates describe a point by how far it sits horizontally (x) and vertically (y) from the origin. Polar coordinates instead describe the same point by its straight-line distance from the origin (r, the radius) and the angle that line makes with the positive x-axis (θ, theta).
The two systems are fully interchangeable: any point that can be written as (x, y) can equally be written as (r, θ), and vice versa. The choice of system depends entirely on the problem. Circular motion, waves, and E-field diagrams are far simpler in polar; linear equations and vector addition are simpler in rectangular.
This converter handles both directions instantly, shows the step-by-step working for each formula, and draws the result on a labelled coordinate plane so the geometry is always clear.
How the Coordinates Relate
The diagram below shows the right-triangle relationship that connects the two systems. The radius vector r is the hypotenuse; x and y are the two legs.
x-component (horizontal leg)
x = r · cos θ
Adjacent side of the right triangle divided by the hypotenuse equals cos θ. Rearranged: x = r · cos θ.
y-component (vertical leg)
y = r · sin θ
Opposite side divided by hypotenuse equals sin θ. Rearranged: y = r · sin θ.
Reverse direction (Pythagorean theorem)
r = √(x² + y²)
θ = atan2(y, x)
atan2(y, x) correctly identifies the angle in all four quadrants using both signs of x and y, unlike plain arctan(y/x) which loses quadrant information.
Complete Conversion Formulas
Polar (r, θ) → Rectangular (x, y)
θ can be in any unit — convert to radians first if needed: θ_rad = θ_deg × π/180
Rectangular (x, y) → Polar (r, θ)
atan2 returns θ ∈ (−π, π] in radians, or (−180°, 180°] in degrees.
Key identity: x² + y² = r²
This is just the Pythagorean theorem applied to the right triangle formed by x, y, and r. It also means cos²(θ) + sin²(θ) = 1, the fundamental trigonometric identity.
How to Convert — Worked Examples
Example 1: Polar (5, 53.13°) → Rectangular
- 1
Convert θ to radians: 53.13° × π/180 = 0.9274 rad - 2
x = r · cos(θ) = 5 · cos(0.9274) = 5 × 0.6 = 3 - 3
y = r · sin(θ) = 5 · sin(0.9274) = 5 × 0.8 = 4 - ✓
Result: (x, y) = (3, 4)
Example 2: Rectangular (−3, 4) → Polar (2nd Quadrant)
- 1
r = √(x² + y²) = √((−3)² + 4²) = √(9 + 16) = √25 = 5 - 2
θ = atan2(4, −3) = 2.2143 rad (NOT arctan(4/−3) = −0.9273) - 3
Convert: 2.2143 × 180/π = 126.87° - ✓
Result: (r, θ) = (5, 126.87°) — Quadrant II ✓
Plain arctan(y/x) = arctan(4/−3) ≈ −53.13° is completely wrong for this case. atan2 handles negative x correctly by examining both signs.
Common Angles Reference (r = 1)
When r = 1 these are just the unit circle values: x = cos(θ), y = sin(θ). Memorising a few key angles makes polar conversions much faster.
| θ (degrees) | θ (radians) | x = cos θ | y = sin θ | Quadrant |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Positive x-axis |
| 30° | π/6 | √3/2 ≈ 0.866 | 1/2 = 0.5 | I |
| 45° | π/4 | 1/√2 ≈ 0.707 | 1/√2 ≈ 0.707 | I |
| 60° | π/3 | 1/2 = 0.5 | √3/2 ≈ 0.866 | I |
| 90° | π/2 | 0 | 1 | Positive y-axis |
| 120° | 2π/3 | −1/2 = −0.5 | √3/2 ≈ 0.866 | II |
| 135° | 3π/4 | −1/√2 ≈ −0.707 | 1/√2 ≈ 0.707 | II |
| 150° | 5π/6 | −√3/2 ≈ −0.866 | 1/2 = 0.5 | II |
| 180° | π | −1 | 0 | Negative x-axis |
| 210° | 7π/6 | −√3/2 ≈ −0.866 | −1/2 = −0.5 | III |
| 225° | 5π/4 | −1/√2 ≈ −0.707 | −1/√2 ≈ −0.707 | III |
| 240° | 4π/3 | −1/2 = −0.5 | −√3/2 ≈ −0.866 | III |
| 270° | 3π/2 | 0 | −1 | Negative y-axis |
| 315° | 7π/4 | 1/√2 ≈ 0.707 | −1/√2 ≈ −0.707 | IV |
| 360° | 2π | 1 | 0 | Positive x-axis |
When to Use Each Coordinate System
Use Polar (r, θ) when…
- •Describing circular or radial motion (orbits, spinning objects)
- •Working with complex numbers in exponential form (r·e^iθ)
- •Drawing spirals, rose curves, limaçons and other radial shapes
- •Expressing electric and magnetic field lines around a point source
- •Integrating over circular or annular regions (polar double integrals)
Use Rectangular (x, y) when…
- •Solving linear equations and systems (slope-intercept, matrix methods)
- •Adding or subtracting vectors component-by-component
- •Working with parabolas, hyperbolas, ellipses in standard form
- •Applying translations (moving a point left/right/up/down)
- •Performing matrix transformations (rotation, scaling, shearing)
Frequently Asked Questions
What is the difference between polar and Cartesian coordinates?
Cartesian (rectangular) coordinates locate a point by its signed horizontal distance x and signed vertical distance y from the origin. Polar coordinates locate the same point by its distance r from the origin (always ≥ 0) and the angle θ the line O→P makes with the positive x-axis. Both systems describe the same 2D plane — they just use different reference measurements. The two systems are linked by x = r·cos θ, y = r·sin θ, r = √(x²+y²), and θ = atan2(y, x).
Why does this converter use atan2 instead of plain arctan?
Plain arctan(y/x) only returns angles in the range (−90°, 90°) and is undefined when x = 0. It also cannot distinguish between (3, 4) and (−3, −4) — both give the same arctan result. atan2(y, x) uses both the sign of y and the sign of x to return the correct angle in all four quadrants, producing a unique result in (−180°, 180°]. For example, atan2(1, −1) = 135° correctly identifies Quadrant II, while arctan(1/−1) = −45° would be wrong.
Can r be negative in polar coordinates?
Some textbooks define (−r, θ) as the point diametrically opposite, i.e. the same as (r, θ + 180°). This converter uses the standard convention r ≥ 0 — consistent with the definition r = √(x²+y²), which is always a non-negative distance. If you have a negative r from a textbook context, add 180° to θ and negate r to get the equivalent standard polar form.
What angle range does this converter output for Rectangular → Polar?
Go's math.Atan2 returns values in the range (−π, π] radians, which in degrees is (−180°, 180°]. Points in Quadrant I give θ ∈ (0°, 90°), Quadrant II give θ ∈ (90°, 180°), Quadrant III give θ ∈ (−180°, −90°), and Quadrant IV give θ ∈ (−90°, 0°). You can always add 360° to a negative result if you prefer the range [0°, 360°).
How do I convert polar coordinates given in radians?
Select 'Radians' from the angle unit toggle before entering your values. The converter applies the formulas directly using your radian input: x = r·cos(θ_rad) and y = r·sin(θ_rad) without any unit conversion step. When going the other direction (Rectangular → Polar) with radians selected, the output θ is also in radians.
What happens if I enter x = 0 and y = 0?
The origin (0, 0) has r = 0 and no defined angle θ — every direction leads to the origin, so θ is ambiguous. This converter still returns a successful result with r = 0 and notes that θ is undefined, rather than showing an error. The coordinate plot shows just the origin dot with the label 'r = 0'.
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