How Cartesian and polar coordinates convert
Use either direction
Choose Cartesian → polar for x and y, or Polar → Cartesian for r and θ. Select degrees or radians before converting. Editing any input or unit clears the old result so it cannot be mistaken for the new values.
The pole and Cartesian origin are the same point, and θ is measured counterclockwise from the positive x-axis.
Conversion formulas
From Cartesian coordinates, r = √(x² + y²) and θ = atan2(y, x). atan2 uses both signs, so points on axes and in all four quadrants receive the correct principal angle.
From polar coordinates, x = r cos θ and y = r sin θ. Degrees are converted to radians internally before sine and cosine are evaluated.
Range and limitations
The output convention is (−180°, 180°] or (−π, π]. Polar input may use negative or multi-turn angles because coterminal angles identify the same point.
This page is for 2D coordinates only. It does not convert spherical, cylindrical, geographic or map coordinates. Ordinary output uses at most two decimals while internal browser precision is retained.
r = √(x² + y²) · θ = atan2(y, x) · x = r cos θ · y = r sin θ
Worked examples
Quadrant I: (3, 4)
r = √(3² + 4²) = 5 and θ = atan2(4, 3) ≈ 53.13°. The reverse conversion returns x = 3 and y = 4.
Axes and negative angles
(−2, 0) becomes (2, 180°). Polar (2, −90°) becomes (0, −2). These checks show why the quadrant-aware atan2 convention matters.
Frequently asked questions
Why is θ undefined at (0, 0)?
The origin has no direction: every angle with radius zero names the same point.
Can I enter a negative angle?
Yes. Negative angles rotate clockwise, and the resulting Cartesian point is valid.
Why can two polar pairs name the same point?
Angles that differ by a full turn are coterminal, such as 0° and 360°.