Cartesian to Polar Calculator

Convert Cartesian coordinates (x, y) to polar coordinates (r, θ) with results in both degrees and radians.

Enter Coordinates

Result

Polar Coordinates
(5, 53.13°)
r = 5, θ = 53.13°
r (radius) 5
θ (degrees) 53.1301°
θ (radians) 0.9273 rad
Quadrant I
x2 + y2 25

Step-by-Step Solution

r = sqrt(3^2 + 4^2) = sqrt(25) = 5

Understanding Coordinate Systems

Two of the most commonly used coordinate systems in mathematics are the Cartesian (rectangular) coordinate system and the polar coordinate system. While the Cartesian system uses two perpendicular axes (x and y) to describe a point's position, the polar system uses a distance from the origin (r) and an angle from the positive x-axis (θ).

Converting between these two systems is a fundamental skill in mathematics, physics, and engineering. Many problems that are difficult in one coordinate system become simple in the other. For example, circles and spirals are naturally described in polar coordinates, while straight lines and rectangles are more natural in Cartesian coordinates.

Conversion Formulas

Cartesian to Polar

Convert rectangular (x, y) coordinates to polar (r, θ) coordinates using the Pythagorean theorem and inverse tangent.

r = sqrt(x^2 + y^2)
θ = atan2(y, x)

Polar to Cartesian

Convert polar (r, θ) coordinates to rectangular (x, y) coordinates using cosine and sine projections.

x = r * cos(θ)
y = r * sin(θ)

Angle Conversion

Convert between degrees and radians, the two most common angular units.

radians = degrees * pi / 180
degrees = radians * 180 / pi

The atan2 Function

The standard arctangent function (atan or tan-1) only returns values in the range (-90°, 90°), which means it cannot distinguish between points in different quadrants that have the same tangent ratio. The atan2(y, x) function solves this problem by considering the signs of both x and y to determine the correct quadrant.

The atan2 function returns an angle in the range (-180°, 180°] or (-π, π] radians. This gives the correct angle for all four quadrants:

  • Quadrant I (x > 0, y > 0): θ is between 0° and 90°
  • Quadrant II (x < 0, y > 0): θ is between 90° and 180°
  • Quadrant III (x < 0, y < 0): θ is between -180° and -90°
  • Quadrant IV (x > 0, y < 0): θ is between -90° and 0°

Practical Applications

Physics and Engineering

Polar coordinates are essential for describing circular motion, orbital mechanics, electromagnetic fields, and wave propagation. For example, the electric field of a point charge is most naturally expressed in polar (or spherical) coordinates because it has radial symmetry.

Navigation and GPS

Navigation systems often use polar-like coordinates. Radar systems measure distance and bearing (angle), which are polar coordinates. Converting these to Cartesian coordinates allows plotting on maps and integrating with other position data.

Computer Graphics

In graphics programming, converting between coordinate systems is essential for drawing arcs, circles, spirals, and implementing rotation transformations. The conversion formulas are used extensively in 2D and 3D rendering engines.

Signal Processing

Complex numbers in signal processing are often represented in both rectangular (real + imaginary) and polar (magnitude + phase) forms. The conversion between these forms is identical to Cartesian-to-polar conversion.

Special Cases

  • Origin (0, 0): r = 0 and θ is undefined (any angle points to the origin).
  • Positive x-axis: When y = 0 and x > 0, θ = 0°.
  • Positive y-axis: When x = 0 and y > 0, θ = 90°.
  • Negative x-axis: When y = 0 and x < 0, θ = 180° (or -180°).

Tips for Coordinate Conversion

  • Always verify your answer by converting back to the original coordinate system.
  • Be careful with the quadrant -- simple arctan may give the wrong angle.
  • Remember that r is always non-negative in standard polar coordinates.
  • When working with complex numbers, use the same conversion: magnitude = r, argument = θ.
  • Degrees are more intuitive for most people, but radians are required for calculus and most formulas.