What this calculator does
Switch a point between its across-and-up position (x, y) and its distance-and-direction position (r, θ). These are the two common ways to say where a point sits on a flat grid. Cartesian (also called rectangular) coordinates say how far across and how far up it is, as (x, y). Polar coordinates say how far it is from the origin, the center point (0, 0), and in which direction. That's a distance r and an angle θ (the Greek letter theta). The angle is measured counterclockwise from the positive x-axis, the line pointing right.
Pick a direction and enter the coordinates you have. The converter gives you the other pair. Going to polar, it shows the angle from −180° to 180° and from 0° to 360°, plus radians (another angle unit). It always places the point in the correct quadrant, one of the four quarters of the grid. Going to Cartesian, you can enter the angle in degrees or radians.
Where it's used
Why this math matters outside the classroom:
- Radar contacts. A radar contact 3 km east and 4 km north of the station shows up at a range of 5 km and about 53.13° from east.
- Robot arms. A robot arm reaching a point at (0.3 m, 0.4 m) must extend 0.5 m and rotate about 53.1° from its base axis.
- Graphics and games. Game developers place an object 10 units from the center at 30° by computing x = 10 cos 30° ≈ 8.66 and y = 10 sin 30° = 5.
- Boat navigation. A boat that sails 12 km on a heading 45° from east ends up about 8.49 km east and 8.49 km north of the harbor.
- Site surveys. A surveyor who measures a corner 50 m away at 120° records it as x = −25 m and y ≈ 43.3 m on the site grid.
How to use it
- Choose Cartesian to polar or polar to Cartesian.
- Enter x and y. Or enter r and θ and pick the angle unit.
- Read the converted point at the top. The separate values and the point's quadrant are listed below it.
What the inputs mean
- x-coordinate
- How far across: positive to the right of the center point, negative to the left.
- y-coordinate
- How far up: positive above the center point, negative below.
- Radius (r)
- The distance from the origin.
- Angle (θ)
- Turning counterclockwise from the line pointing right. Negative angles turn clockwise. Adding 360° gives the same direction.
Formula and method
r = √(x² + y²) θ = atan2(y, x) x = r × cos θ y = r × sin θ
Here x and y are the across and up positions, r is the distance from the center and θ is the angle. The distance comes from the Pythagorean theorem, because x, y and r form a right triangle. The angle isn't just arctan(y ÷ x), the inverse tangent. That ratio is the same for opposite points such as (1, 1) and (−1, −1), and it breaks when x is 0. The two-input version, atan2(y, x), checks the signs of both coordinates. So it returns the angle in the correct quadrant, between −180° and 180°.
Going the other way, the cosine and sine of the angle give the across and up shares of the distance. At the special angles (multiples of 30° and 45°), the calculator uses exact sine and cosine values. So 90° gives x = 0 exactly, not a tiny leftover.
Worked example
A drone is 6 m east and 8 m south of its launch point, so it sits at (6, −8). What are its polar coordinates?
- Radius: r = √(6² + (−8)²) = √100 = 10 m.
- Angle: θ = atan2(−8, 6) ≈ −53.130102°. That's below the positive x-axis, in quadrant IV (the lower-right quarter).
- As a positive angle, that's −53.130102° + 360° = 306.869898°, or about −0.927295 radians.
Result: The drone is at (10, −53.130102°) in polar coordinates. That's 10 m from launch, about 53.13° clockwise from due east.
Understanding the result
Both angle ranges describe the same direction. The −180° to 180° form is common in programming and navigation. The 0° to 360° form is the usual choice in math class.
A point has more than one polar form. (r, θ) and (r, θ + 360°) are the same point, and the origin has r = 0 with any angle. This converter shows the standard form, with r at 0 or above.
Assumptions and limitations
- Angles are measured counterclockwise from the positive x-axis.
- The radius you enter for polar input is zero or positive.
- It handles only flat, two-dimensional points. Cylindrical and spherical coordinates (for 3D space) aren't included.
- Negative radii, which some textbooks allow, aren't accepted. Add 180° to the angle instead.
Frequently asked questions
Why not just use arctan(y ÷ x)?
Because it only returns angles between −90° and 90°. The points (3, 4) and (−3, −4) give the same ratio, but they point in opposite directions. atan2 looks at the signs of x and y separately, so it gets the quadrant right. It also works when x is 0.
What are the polar coordinates of the origin?
The radius is 0, and the angle can be anything, since every direction leads nowhere. By custom, the converter shows θ = 0.
Are rectangular and Cartesian coordinates the same thing?
Yes. Both names mean the (x, y) system built on two axes at right angles. Rectangular is often used when talking about complex numbers.
How do I convert degrees to radians?
Multiply by π ÷ 180. For example, 90° is π ÷ 2 ≈ 1.570796 radians. The converter shows the angle in both units.
How this calculator is checked
Method: r = √(x² + y²), θ = atan2(y, x), x = r cos θ, y = r sin θ. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
Embed this calculator
Put this Cartesian to Polar Converter on your own website or blog for free. Paste this code where you want it to appear: