Rectangular to Polar Coordinates Converter

Convert rectangular (Cartesian) coordinates (x, y) to polar coordinates (r, θ). Essential tool for mathematics, physics, and engineering applications.

Conversion Formulas:

From rectangular (x, y) to polar (r, θ):

r = √(x² + y²)      θ = atan2(y, x)

Note: atan2(y, x) gives the correct quadrant for the angle θ

Horizontal distance from the y-axis
Vertical distance from the x-axis
Unit for the angle θ in the polar coordinates result
Range of x and y values to display on the graph
II
( -, + )
I
( +, + )
III
( -, - )
IV
( +, - )
The atan2(y, x) function automatically determines the correct quadrant based on the signs of x and y
(3.54, 3.54)
(0, 5)
(-4, 0)
(0, -3)
(5.20, 3)
(0.5, 0.87)
(-2, -2)
(1, -1.73)
Calculating...

Understanding Rectangular and Polar Coordinates

Coordinate systems are essential in mathematics, physics, engineering, and many other fields for describing the position of points in space.

Rectangular (Cartesian) Coordinates (x, y):

  • x (abscissa): Horizontal distance from the y-axis (positive to the right, negative to the left)
  • y (ordinate): Vertical distance from the x-axis (positive upward, negative downward)
  • Points are located at the intersection of perpendicular grid lines
  • Standard coordinate system in most mathematical applications

Polar Coordinates (r, θ):

  • r (radius): Distance from the origin (pole) to the point (always r ≥ 0)
  • θ (angle): Angle measured counterclockwise from the positive x-axis (polar axis)
  • Angle can be measured in degrees, radians, or gradians
  • Multiple representations: (r, θ + 360°k) or (r, θ + 2πk) for any integer k

Conversion Formulas

Conversion Formulas Notes
Rectangular to Polar r = √(x² + y²)
θ = atan2(y, x) 		atan2 gives correct quadrant for θ (0 to 2π)
Polar to Rectangular x = r·cos(θ)
y = r·sin(θ) 		Direct conversion using trigonometric functions
Quadrant Determination I: x>0, y>0 → 0° < θ < 90°
II: x<0, y>0 → 90° < θ < 180°
III: x<0, y<0 → 180° < θ < 270°
IV: x>0, y<0 → 270° < θ < 360° 		atan2(y, x) automatically handles this
Special Cases Origin: (0, 0) → r=0, θ undefined
On x-axis: y=0 → θ=0° (x>0) or 180° (x<0)
On y-axis: x=0 → θ=90° (y>0) or 270° (y<0) 		For origin, θ is typically taken as 0 by convention

The atan2(y, x) Function

The atan2(y, x) function is essential for correct rectangular to polar conversion because it considers the signs of both x and y to determine the correct quadrant for the angle θ.

1

Why not use arctan(y/x)? The regular arctan(y/x) only gives angles in the range (-90°, 90°), which covers only Quadrants I and IV. It cannot distinguish between points in Quadrants II and III.

2

How atan2 works: atan2(y, x) returns the angle from the positive x-axis to the point (x, y), with the result in the range (-π, π] radians or (-180°, 180°] degrees.

3

Standard conversion: To get θ in the standard range [0, 2π) or [0°, 360°), add 2π or 360° to negative angles.

Applications of Coordinate Conversion

1

Physics: Converting position vectors from Cartesian to polar form for problems with circular or rotational symmetry.

2

Engineering: Signal processing where polar form (magnitude and phase) is more convenient than rectangular form (real and imaginary parts).

3

Computer Graphics: Converting between coordinate systems for rotations, transformations, and texture mapping.

4

Navigation: Converting Cartesian map coordinates to bearing and distance for navigation purposes.

Common Coordinate Examples

Origin
(0, 0) → (0, 0°)
Zero radius, angle arbitrary
On x-axis
(5, 0) → (5, 0°)
Positive x-axis
On y-axis
(0, 3) → (3, 90°)
Positive y-axis
Quadrant I
(3, 4) → (5, 53.13°)
3-4-5 triangle
Quadrant II
(-1, 1) → (√2, 135°)
45° from negative x-axis
Quadrant III
(-2, -2) → (2√2, 225°)
45° from negative x-axis
Quadrant IV
(1, -√3) → (2, 300°)
60° from positive x-axis
Unit circle
(cosθ, sinθ) → (1, θ)
Any point on unit circle

Calculator Features:

  • Convert rectangular coordinates to polar coordinates with precision
  • Support for degrees, radians, and gradians angle units
  • Step-by-step calculation showing all intermediate steps
  • Visual representation of both coordinate systems
  • Angle visualization with interactive quadrant display
  • Example coordinates for quick testing
  • Automatic quadrant determination using atan2 function

Frequently Asked Questions

The atan2(y, x) function is the two-argument arctangent function that computes the angle from the positive x-axis to the point (x, y). Unlike the regular arctan(y/x) which only gives angles between -90° and 90°, atan2 considers the signs of both x and y to determine the correct quadrant, giving angles in the range (-180°, 180°] or [0°, 360°) after adjustment. This is crucial for correct rectangular to polar conversion.

For the origin (0, 0), the radius r = 0. The angle θ is technically undefined because there's no unique direction from the origin to itself. By convention, we usually take θ = 0° (or 0 radians) for the origin. Some applications may treat it as undefined or assign it a special value. This calculator follows the convention of θ = 0° for the origin.

In standard polar coordinates, the radius r is always non-negative (r ≥ 0). However, some conventions allow negative r values, which are interpreted as the point being in the opposite direction of the angle θ. Mathematically, (-r, θ) is equivalent to (r, θ + 180°). This calculator uses the standard convention with r ≥ 0.

In polar coordinates, adding any multiple of 360° (or 2π radians) to the angle θ gives the same direction. For example, (5, 45°), (5, 405°), and (5, 765°) all represent the same point. Also, as mentioned above, negative r values with angle shifted by 180° can represent the same point. This calculator provides the principal value with r ≥ 0 and 0° ≤ θ < 360°.

This calculator uses JavaScript's built-in Math functions which provide double-precision floating-point arithmetic (about 15-17 decimal digits of precision). For most practical applications, this is more than sufficient. The results are displayed with 4 decimal places by default, but the underlying calculations maintain full precision.