Polar to Cartesian Converter

Convert between polar coordinates (r, theta) and Cartesian coordinates (x, y) with step-by-step solutions.

Enter Polar Coordinates

Results

Cartesian Coordinates
--
x = r cos(theta) --
y = r sin(theta) --
theta (radians) --
theta (degrees) --
Distance from origin --
Quadrant --
Reverse: Cartesian to Polar
r = sqrt(x^2 + y^2) --
theta = atan2(y, x) --

Step-by-Step Solution

Understanding Polar and Cartesian Coordinates

Polar and Cartesian are two different coordinate systems for describing the position of points in a 2D plane. Cartesian coordinates use horizontal (x) and vertical (y) distances from the origin, while polar coordinates use a distance (r) from the origin and an angle (theta) from the positive x-axis.

Conversion Formulas

Polar to Cartesian: x

The horizontal component using cosine.

x = r * cos(theta)

Polar to Cartesian: y

The vertical component using sine.

y = r * sin(theta)

Cartesian to Polar: r

Distance from origin using Pythagorean theorem.

r = sqrt(x^2 + y^2)

Cartesian to Polar: theta

Angle using the two-argument arctangent.

theta = atan2(y, x)

Common Angle Conversions

  • 0 degrees = 0 radians: (r, 0) becomes (r, 0)
  • 30 degrees = pi/6 radians: (r, pi/6) becomes (r*sqrt(3)/2, r/2)
  • 45 degrees = pi/4 radians: (r, pi/4) becomes (r*sqrt(2)/2, r*sqrt(2)/2)
  • 60 degrees = pi/3 radians: (r, pi/3) becomes (r/2, r*sqrt(3)/2)
  • 90 degrees = pi/2 radians: (r, pi/2) becomes (0, r)
  • 180 degrees = pi radians: (r, pi) becomes (-r, 0)

When to Use Each System

Cartesian coordinates are best for problems involving straight lines, rectangles, and grid-based systems. Polar coordinates excel when dealing with circles, spirals, rotational symmetry, and angular motion. Many physics problems (orbital mechanics, wave propagation) are naturally described in polar coordinates.

Applications

  • Navigation: Radar systems use polar coordinates (distance and bearing).
  • Physics: Circular motion, gravitational fields, and electromagnetic fields.
  • Engineering: Antenna patterns, gear design, and CAD drawings.
  • Mathematics: Polar curves like cardioids, roses, and spirals.