Cartesian to Polar Calculator
The point (3, 4) converts to polar form r = 5 at an angle of 53.13 degrees. This Cartesian to polar calculator moves a point between the rectangular (x, y) system and the polar (r, theta) system in either direction. Enter an x and y pair to get the radius, the angle in both degrees and radians, and the quadrant the point sits in, or switch the direction and enter a radius and angle to recover the original x and y values. Every result is rounded for readability while the angle keeps enough precision to convert cleanly back to the point you started with.
Quick answer
The radius r is the straight-line distance from the origin to the point, found with r = sqrt(x squared + y squared).
What this tells you
- •The radius r is the straight-line distance from the origin to the point, found with r = sqrt(x squared + y squared).
- •The angle theta is measured from the positive x-axis, found with the atan2 function so the correct quadrant is always used.
- •Degrees are normalized to a 0 to 360 range, so a point below the x-axis reports an angle like 315 rather than a negative value.
- •Radians are reported as the raw atan2 result between negative pi and pi, which is the form most calculus and physics work expects.
- •Switching to polar to Cartesian mode reverses the process with x = r times cos(theta) and y = r times sin(theta).
- •The radius must be zero or greater, matching the standard convention that a polar radius is never negative.
How to Use
- 11. Choose the conversion direction. Leave it on Cartesian to polar to turn an (x, y) point into r and theta, or switch to Polar to Cartesian to go the other way.
- 22. In Cartesian to polar mode, enter the x value and the y value of your point. Both can be positive, negative, or zero.
- 33. In Polar to Cartesian mode, enter the radius r and the angle theta in degrees. The radius must be zero or greater.
- 44. Read the primary result, then check the secondary values for the angle in radians and the quadrant so you can confirm the point landed where you expect.
- 55. To verify a round trip, convert a point to polar, then paste the radius and degree angle back in Polar to Cartesian mode and confirm you recover the original coordinates.
How It Works
Formula
r = sqrt(x^2 + y^2), theta = atan2(y, x). x = r cos(theta), y = r sin(theta)Converting a rectangular point to polar form uses the Pythagorean distance for the radius and the atan2 function for the angle. The atan2 function takes both x and y as separate arguments, which lets it place the angle in the correct quadrant instead of losing sign information the way a plain arctangent of y divided by x would. The calculator reports the degree angle on a 0 to 360 scale by adding 360 to any negative raw angle, while the radian value stays in the natural negative pi to pi range. Going the other direction is direct trigonometry, where x is the radius times the cosine of the angle and y is the radius times the sine of the angle, with the angle converted from degrees to radians internally.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
A 3-4-5 point in quadrant I
The radius is the square root of 3 squared plus 4 squared, which is the square root of 25, or exactly 5. The angle atan2(4, 3) is about 0.9273 radians, which is 53.13 degrees, placing the point in the first quadrant.
A point in quadrant III with a normalized angle
The radius is still 5 because squaring removes the signs. The raw atan2 angle is negative, so the tool adds 360 and reports 233.13 degrees, which correctly points into the third quadrant instead of showing a confusing negative value.
A point on the negative y-axis
With x equal to 0 and y negative, the point sits straight down from the origin. The radius equals the absolute value of y, which is 6, and the angle is 270 degrees. The quadrant label reads Negative y-axis rather than a numbered quadrant.
Polar back to Cartesian
Feeding the radius 5 and angle 53.1301 degrees back into the reverse mode returns x equal to 5 times cos(53.1301 degrees), about 3, and y equal to 5 times sin(53.1301 degrees), about 4, recovering the point the first example started from.
Common Cartesian to Polar Conversions
Sample points and their polar coordinates using a 0 to 360 degree angle range.
| Point (x, y) | Radius r | Angle (degrees) | Quadrant |
|---|---|---|---|
| (1, 0) | 1 | 0 | Positive x-axis |
| (1, 1) | 1.4142 | 45 | Quadrant I |
| (0, 1) | 1 | 90 | Positive y-axis |
| (-1, 1) | 1.4142 | 135 | Quadrant II |
| (-1, -1) | 1.4142 | 225 | Quadrant III |
| (1, -1) | 1.4142 | 315 | Quadrant IV |
Radius values are rounded to four decimals. A point at the origin has a radius of 0 and no defined angle.
Why atan2 beats a plain arctangent
The angle of a polar coordinate is easy to get wrong if you compute it as the arctangent of y divided by x. That formula throws away the individual signs of x and y and only keeps the sign of their ratio, so the points (3, 4) and (-3, -4) produce the same arctangent even though they sit on opposite sides of the plane. You would then have to remember to add or subtract 180 degrees by hand based on which quadrant the point belongs to.
The atan2 function solves this by taking x and y as two separate inputs. Because it can see both signs, it returns the angle in the correct quadrant on its own, across the full negative pi to pi range. This calculator uses atan2 and then shifts any negative result up by 360 degrees so the reported angle always reads as a positive value between 0 and 360, which is the range most homework problems and reference charts expect.
Keeping the radian value in its raw negative pi to pi form is deliberate. Calculus, physics, and programming work usually expect radians in that range, so the tool shows both the tidy 0 to 360 degree value for reading and the raw radian value for further math.
Common mistakes
- Using arctangent of y over x instead of atan2, which loses quadrant information and can put the angle 180 degrees away from the true direction.
- Mixing degrees and radians. This tool takes and returns the degree angle on a 0 to 360 scale and also shows radians, so confirm which unit your assignment or downstream formula expects.
- Entering a negative radius in Polar to Cartesian mode. A polar radius is zero or greater by convention, so the tool rejects negative values rather than guessing your intent.
- Assuming the origin has an angle. When both x and y are 0 the radius is 0 and the angle is undefined, so the tool labels the point as the origin instead of reporting a false angle.
- Forgetting that the angle is measured from the positive x-axis counterclockwise, not from the positive y-axis or clockwise, which flips or rotates the result.
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