Polar to Rectangular Coordinate Calculator
This calculator converts between polar coordinates (r, theta) and rectangular (Cartesian) coordinates (x, y) in both directions. In polar form, r is the distance from the origin and theta is the angle measured counterclockwise from the positive x-axis. To convert from polar to rectangular: x = r times cos(theta) and y = r times sin(theta). To convert from rectangular to polar: r = sqrt(x squared + y squared) and theta = atan2(y, x). The atan2 function correctly identifies the quadrant of the point, unlike the basic arctan which only returns values in the range -90 to 90 degrees. Angles can be entered in degrees or radians. Polar coordinates are especially useful for describing circles, spirals, and curves with rotational symmetry. For example, the circle r = 5 corresponds to all points exactly 5 units from the origin, which in rectangular coordinates is x squared + y squared = 25.
Conversion formulas
Polar (r, theta) to Rectangular (x, y):
x = r * cos(theta)
y = r * sin(theta)
Rectangular (x, y) to Polar (r, theta):
r = sqrt(x^2 + y^2)
theta = atan2(y, x) [gives correct quadrant]
Reference: common angle conversions
| Degrees | Radians | cos(theta) | sin(theta) |
|---|---|---|---|
| 0° | 0 | 1.00 | 0.00 |
| 30° | pi/6 | 0.866 | 0.500 |
| 45° | pi/4 | 0.707 | 0.707 |
| 60° | pi/3 | 0.500 | 0.866 |
| 90° | pi/2 | 0.00 | 1.00 |
| 180° | pi | -1.00 | 0.00 |
| 270° | 3pi/2 | 0.00 | -1.00 |
Polar coordinates: frequently asked questions
What are polar coordinates?
Polar coordinates represent a point in the plane by its distance r from the origin (the pole) and the angle theta measured counterclockwise from the positive x-axis (the polar axis). Every point (x,y) in the plane has a polar representation (r, theta). Polar coordinates are especially useful for curves with circular symmetry.
How do you convert polar (r, theta) to rectangular (x, y)?
Use the formulas x = r * cos(theta) and y = r * sin(theta), where theta is in radians (or convert degrees to radians first by multiplying by pi/180). For example, (r=2, theta=30°) gives x = 2*cos(30°) = 2*(sqrt(3)/2) = sqrt(3) approximately 1.732, and y = 2*sin(30°) = 2*(0.5) = 1.
How do you convert rectangular (x, y) to polar (r, theta)?
Compute r = sqrt(x^2 + y^2). Compute theta = atan2(y, x), which gives the angle in the correct quadrant. The atan2 function returns values in (-pi, pi] radians. Convert to degrees by multiplying by 180/pi. For (x=1, y=1): r = sqrt(2) approximately 1.414 and theta = 45°.
Why use atan2 instead of atan(y/x)?
The standard arctangent atan(y/x) only returns values in the range (-90°, 90°), so it cannot distinguish between points in opposite quadrants. For example, (1,1) and (-1,-1) both give atan(1/1) = 45°, but they are in opposite quadrants. The atan2(y,x) function uses both the sign of x and y to return the correct angle in all four quadrants.
Can r be negative in polar coordinates?
Conventionally, r is taken to be non-negative. A negative r means the point is in the opposite direction: (-r, theta) = (r, theta + 180°). Some contexts allow negative r, but this calculator converts negative r to positive r with the appropriate angle adjustment.
Sources
- Coordinate systems: NIST DLMF 1.5, Coordinate Systems.
- Polar coordinates: Wolfram MathWorld, Polar Coordinates.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.