Cartesian to Polar Coordinates
Enter a point (x, y) in the Cartesian plane. The calculator computes the polar coordinates (r, θ) using r = √(x² + y²) and θ = atan2(y, x), reporting θ in both degrees and radians, normalized to [0°, 360°).
From coordinates to distance and direction
Cartesian coordinates locate a point by how far to go along two perpendicular axes. Polar coordinates locate the same point by how far it is from the origin and in what direction. Neither is more correct; each makes a different kind of problem easy.
This calculator converts a point (x, y) into its polar form (r, θ), reporting the angle in both degrees and radians and normalising it to the range 0° up to but not including 360°.
Circular and rotational problems are usually clumsy in Cartesian form and clean in polar form. A circle is x² + y² = 25 in one and simply r = 5 in the other, and anything defined by distance from a centre follows the same pattern.
The conversion also arises whenever a vector given by components is needed as a magnitude and a bearing: a velocity given as 3 east and 4 north is the same thing as a speed of 5 in a particular direction, and this tool performs that translation.
How to use this calculator
- Enter the x coordinate The horizontal displacement from the origin, negative to the left. Any real number is accepted, including zero.
- Enter the y coordinate The vertical displacement, negative downwards. The signs of x and y together decide the quadrant, which is what makes the angle unambiguous.
- Read r, the radius The straight-line distance from the origin to the point. It is never negative, whatever the signs of the inputs.
- Read θ in whichever unit you need Both are given. The degree value is normalised into [0°, 360°), so a point below the x-axis is reported with a large positive angle rather than a negative one.
The formula, and where it comes from
r = √(x² + y²) θ = atan2(y, x), normalised to [0°, 360°)
The radius is the Pythagorean theorem applied to the two components: the point, the origin and the foot of the perpendicular form a right triangle whose hypotenuse is r. Squaring removes the signs, which is why r is always non-negative.
The angle uses atan2 rather than the arctangent of y/x, and the distinction matters. Dividing y by x throws away which of the two was negative, so (1, 1) and (−1, −1) produce the same quotient despite pointing in opposite directions, and the division is undefined when x is zero. atan2 takes both components separately, so it resolves the quadrant correctly and handles the vertical axis without a special case.
What each input means
- x Horizontal coordinate — form field “x”
- Signed distance along the horizontal axis. Its sign, combined with that of y, determines which quadrant the point lies in and therefore which angle atan2 returns.
- y Vertical coordinate — form field “y”
- Signed distance along the vertical axis, positive upwards in the standard orientation.
- r Radius
- The distance from the origin, always zero or positive. It is the magnitude of the vector from the origin to the point.
- θ Polar angle
- The direction, measured anticlockwise from the positive x-axis and reported in [0°, 360°). It is the same quantity as the argument of a complex number x + iy. Units: degrees and radians, both reported.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
The 3-4-5 point
Convert (3, 4). The components form the best-known Pythagorean triple, so the radius should come out exactly, which makes this a good check that the conversion is behaving.
Inputs x = 3, y = 4
- Cartesian point (3, 4)
- Radius r = √(x² + y²) = √(9 + 16) = 5
- Angle θ = atan2(y, x) = 53.1301° = 0.927295 rad
Result (r, θ) = (5, 53.1301°) = (5, 0.927295 rad)
The radius is exactly 5, because 3² + 4² = 25. The angle is a little under 54°, placing the point in the first quadrant — consistent with both coordinates being positive.
The angle is not one of the memorable values, which is normal: only points where the ratio of the coordinates matches a special triangle produce angles like 30° or 45°. A decimal angle is the usual case, not a sign of a problem.
A point on the negative y-axis
Convert (0, −3). This is the case where dividing y by x would fail outright, so it demonstrates why atan2 is used.
Inputs x = 0, y = -3
- Cartesian point (0, -3)
- Radius r = √(x² + y²) = √(0 + 9) = 3
- Angle θ = atan2(y, x) = 270° = 4.71239 rad
Result (r, θ) = (3, 270°) = (3, 4.71239 rad)
The radius is 3 and the angle is 270°, pointing straight down. Had the tool computed arctan(y/x) it would have divided by zero; atan2 handles a zero x directly and returns the correct quarter turn.
Note that the angle is reported as 270° rather than −90°. The two describe the same direction, but the tool normalises to a non-negative range, so subtract 360° if your work uses the (−180°, 180°] convention instead.
Reading the result
The angle convention in use
θ is measured anticlockwise from the positive x-axis and reported in [0°, 360°). Many textbooks and most programming libraries return atan2 in (−180°, 180°] instead. The two differ by a full turn for points below the axis, and both are correct — but mixing them within one calculation is not.
The origin is a special case
At (0, 0) the radius is zero and the direction is genuinely undefined, since no ray is picked out. The tool reports the angle as 0 by convention. That is a convenient choice rather than a mathematical fact, and it should not be read as meaning the point lies along the positive x-axis.
Precision of the reported values
Both r and θ are shown to six significant figures. Values such as the 5 above are exact; an angle like 53.1301° is a rounded decimal, and converting it back will reproduce the original coordinates only to that precision.
When you would use this
Turning vector components into magnitude and direction
A force or velocity given by its components converts directly: r is the magnitude and θ the direction. This is the standard first step when combining vectors that were measured along axes but need to be reported as a single push in one direction.
Preparing a curve for polar analysis
Curves with rotational symmetry — circles, cardioids, roses — have far simpler polar equations. Converting sample points is a practical way to recognise which polar form a Cartesian dataset corresponds to.
Assumptions and limitations
What this calculator assumes
- The angle is measured anticlockwise from the positive x-axis, the standard mathematical orientation rather than a compass bearing.
- The radius is reported as non-negative; the alternative convention allowing negative r with the opposite angle is not used.
- The angle is normalised into [0°, 360°), so no reported angle is negative.
- Both coordinates are plain real numbers in the same unit; the tool performs no scaling or unit conversion.
Where it stops being the right tool
- Two dimensions only. Spherical or cylindrical coordinates in three dimensions need a different conversion.
- The reverse direction is not performed here — converting (r, θ) back to (x, y) is a separate solver.
- Compass bearings, measured clockwise from north, are not supported and will be placed in the wrong quadrant if entered directly.
Common mistakes
Using arctan(y/x) instead of atan2
Why it happens. The formula θ = arctan(y/x) is the one printed in most textbooks, and it is correct only in the first quadrant. Elsewhere it loses the sign information and lands the point 180° from where it belongs.
How to avoid it. Compare against the quadrant implied by the signs of your inputs. If the reported angle is in the wrong half of the plane, a plain arctangent is almost always why.
Expecting a negative angle for points below the axis
Why it happens. Programming languages return atan2 in the range (−180°, 180°], so a point in the fourth quadrant is habitually thought of as having a negative angle.
How to avoid it. The two forms differ by exactly 360°. Subtract a full turn from any reported angle above 180° to convert to the signed convention.
Treating r as if it could be negative
Why it happens. Some polar-graphing conventions do allow a negative radius, meaning the point is plotted in the opposite direction, so the idea is not unfamiliar.
How to avoid it. This tool always returns r ≥ 0 and adjusts θ instead. If your source uses negative radii, add 180° to the angle and take the absolute value of r to translate.
Frequently asked questions
Why does the calculator use atan2 rather than arctan?
Because arctan(y/x) discards the individual signs of x and y, so it cannot distinguish opposite quadrants, and it is undefined when x is zero. atan2 takes both components separately, returning the correct angle everywhere including on the vertical axis.
What is the angle at the origin?
Undefined in principle, since the point picks out no direction. The calculator reports 0 by convention. Treat it as a placeholder rather than as a claim that the point lies along the positive x-axis.
Why is θ normalised to [0°, 360°)?
It is the more common convention for Cartesian-to-polar conversion in textbooks. Programming libraries usually return (−180°, 180°] instead; the two agree above the x-axis and differ by a full turn below it, so add or subtract 360° to move between them.
Can the radius come out negative?
No. It is computed as a square root of a sum of squares, so it is always zero or positive. Conventions that permit a negative radius encode direction in the sign instead; this tool keeps r non-negative and puts all the direction information in θ.