Polar Coordinates Calculator
Enter a point in rectangular form (x, y) or in polar form (r, theta) and this calculator converts it to the other system instantly. Choose whether theta is displayed in degrees or radians, and read through the step-by-step working to see exactly how each conversion is done.
What are polar coordinates?
Every point in a plane can be described by two different coordinate systems. The rectangular (or Cartesian) system uses two numbers, x (horizontal distance from the origin) and y (vertical distance). The polar system instead uses r (the straight-line distance from the origin) and theta (the angle measured counter-clockwise from the positive x-axis). Both systems describe the same point, they just use different language to say where it is. Polar coordinates are especially useful in physics and engineering when the problem has circular or rotational symmetry, such as describing the position of a planet in its orbit, the direction of a radar echo, or the path of a spiraling curve.
Rectangular to polar conversion
Given a point (x, y) in rectangular form, the polar radius r is the straight-line distance from the origin: r = sqrt(x^2 + y^2). This is just the Pythagorean theorem applied to the right triangle formed by x, y, and r. The angle theta is found using the two-argument arctangent, commonly written atan2(y, x), which gives the angle in the correct quadrant for all combinations of positive and negative x and y. The result falls in the range (-180, 180] degrees or (-pi, pi] radians. For example, the rectangular point (3, 4) gives r = sqrt(9 + 16) = sqrt(25) = 5, and theta = atan2(4, 3) which is approximately 53.13 degrees.
Polar to rectangular conversion
Given a polar point (r, theta), the rectangular coordinates follow directly from the definitions of cosine and sine: x = r * cos(theta) and y = r * sin(theta). The radius r must be zero or positive. Theta can be any angle, since adding 360 degrees (or 2pi radians) gives the same point. For example, the polar point (5, 53.13 degrees) gives x = 5 * cos(53.13 deg) which is approximately 3, and y = 5 * sin(53.13 deg) which is approximately 4, recovering the original rectangular point (3, 4).
Degrees vs radians and the atan2 function
Radians are the natural unit for angle in mathematics: one radian is the angle that subtends an arc length equal to the radius. A full circle is 2pi radians (approximately 6.2832) or 360 degrees. To convert degrees to radians multiply by pi/180; to convert radians to degrees multiply by 180/pi. The atan2(y, x) function is an extension of the ordinary arctangent that takes both x and y separately rather than their ratio, allowing it to distinguish angles in all four quadrants. Ordinary arctan(y/x) cannot tell whether (1, 1) and (-1, -1) point in opposite directions because both give y/x = 1. Using atan2 eliminates that ambiguity.
Theta angle by quadrant
| x sign | y sign | Quadrant | Theta range (degrees) | Theta range (radians) |
|---|---|---|---|---|
| + | + | I | 0 to 90 | 0 to pi/2 |
| - | + | II | 90 to 180 | pi/2 to pi |
| - | - | III | -180 to -90 | -pi to -pi/2 |
| + | - | IV | -90 to 0 | -pi/2 to 0 |
| 0 | + | Positive y-axis | 90 | pi/2 |
| 0 | - | Negative y-axis | -90 | -pi/2 |
| + | 0 | Positive x-axis | 0 | 0 |
| - | 0 | Negative x-axis | 180 or -180 | pi or -pi |
The sign of x and y determines the quadrant and the principal-value range of theta (using atan2).
Frequently asked questions
What is the formula for converting rectangular to polar coordinates?
The radius is r = sqrt(x^2 + y^2), which is the Pythagorean distance from the origin to the point. The angle is theta = atan2(y, x), which returns the angle in the correct quadrant for all combinations of positive and negative x and y. The result is in radians; multiply by 180/pi to convert to degrees.
What is the formula for converting polar to rectangular coordinates?
The horizontal coordinate is x = r * cos(theta) and the vertical coordinate is y = r * sin(theta). Make sure theta is in radians if your calculator or code expects radians, or convert with theta_rad = theta_deg * pi / 180 first.
Why does theta have a range of (-180, 180] degrees?
The atan2 function returns the principal-value angle, meaning it picks the unique angle in the half-open interval (-pi, pi] radians, which is (-180, 180] degrees. This convention avoids ambiguity: every (x, y) pair maps to exactly one principal theta. You can always add or subtract multiples of 360 degrees to get equivalent representations.
Can r be negative in polar coordinates?
Conventionally r is non-negative. A negative r is sometimes used in advanced graphing (where the point is plotted in the opposite direction at distance |r|), but this calculator follows the standard convention where r >= 0 and the direction is fully determined by theta.
What is the polar form of a point on the axes?
A point on the positive x-axis has theta = 0. A point on the positive y-axis has theta = 90 degrees (pi/2 radians). A point on the negative x-axis has theta = 180 degrees (pi radians). A point on the negative y-axis has theta = -90 degrees (-pi/2 radians). The origin itself has r = 0 and theta is undefined.