Enter Cartesian coordinates (x, y) to get polar coordinates (r, θ), or enter polar coordinates (r, θ) to get Cartesian coordinates (x, y). Choose your conversion direction and the Polar Coordinates Calculator returns the result in both radians and degrees. Also try the Dot Product Calculator.
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Given a Cartesian point (x, y), the polar radius is r = √(x² + y²) and the angle is θ = arctan(y/x). Be sure to account for the quadrant of the point when computing θ — using atan2(y, x) handles all four quadrants correctly and gives angles in the range (−π, π]. See also our use the Vector Addition Calculator.
Given polar coordinates (r, θ), the Cartesian equivalents are x = r × cos(θ) and y = r × sin(θ). Make sure θ is in the correct unit (radians or degrees) before applying the trigonometric functions.
Yes, every point (x, y) in the Cartesian plane has a polar representation (r, θ). The representation is not unique — adding any multiple of 2π to θ gives the same point, and using a negative r with θ + π is also valid. By convention, r ≥ 0 and θ ∈ [0, 2π) or (−π, π] is typically chosen.
The origin (0, 0) in polar form is (0, θ) for any angle θ, since r = 0 regardless of direction. By convention, it is often written as (0, 0), but strictly speaking the angle is undefined at the origin. You might also find our calculate Direction Angle (θ), Direction Angle (Radians) & Magnitude ‖v‖ — Vector Direction useful.
Using x = r cos θ and y = r sin θ: x = 2 × cos(π) = −2 and y = 2 × sin(π) = 0. So the Cartesian point is (−2, 0), which lies on the negative x-axis at a distance of 2 from the origin.
The radius r (also called the radial distance or modulus) represents the straight-line distance from the origin (pole) to the point. It is always non-negative in the standard convention and equals √(x² + y²) when converting from Cartesian coordinates.
The standard arctan function only returns angles in (−π/2, π/2), so it cannot distinguish between points in opposite quadrants (e.g. (1,1) and (−1,−1) give the same arctan value). The atan2(y, x) function uses the signs of both x and y to return the correct angle across all four quadrants in (−π, π].
Standard polar coordinates are a 2D system. For three-dimensional space, the analogues are cylindrical coordinates (r, θ, z) and spherical coordinates (ρ, θ, φ), which extend the polar idea by adding a vertical dimension or a second angle.