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Rectangular to Polar Calculator

r: --

theta: --

How this rectangular to polar calculator works

Enter rectangular coordinates x and y. The calculator finds the polar pair (r, θ). r = √(x² + y²) is the distance from the origin, shown to four decimal places. θ = atan2(y, x) converted to degrees is the angle from the positive x-axis, shown to two decimal places with a degree sign. For x=3, y=4 you get r = 5.0000 and theta = 53.13°.

atan2 keeps the correct quadrant. (1, 0) is 0.00°, (0, 1) is 90.00°, (−1, 0) is 180.00°, and (0, −1) is −90.00°. A point such as (−3, 4) stays in the second quadrant with r = 5.0000 and theta = 126.87°. Using plain atan(y/x) would miss those sign cases when x is negative or zero.

Both fields must hold finite numbers. Negatives and decimals are allowed. An empty box leaves r and theta as “--.” At the origin, r is 0.0000 and theta shows 0.00°. Angles typically fall between −180° and 180°.

To reverse the conversion, use x = r cos(θ) and y = r sin(θ) with the same degree measure converted for trigonometry. Check r by squaring x and y, adding, and taking the square root—the same magnitude as a vector length.

Rectangular to polar examples

Each row matches √(x² + y²) and atan2(y, x) in degrees as on the calculator.

xyrthetaWhat it marks
345.000053.13°First quadrant
51213.000067.38°5–12–13 magnitude
111.414245.00°45° line
101.00000.00°Positive x-axis
011.000090.00°Positive y-axis
−101.0000180.00°Negative x-axis
0−11.0000−90.00°Negative y-axis
−345.0000126.87°Second quadrant

How to read the result

  • r = √(x² + y²)

    Distance from the origin; 3, 4 → 5.0000.

  • theta from atan2

    Angle in degrees; (1, 1) → 45.00°.

  • Correct quadrant

    (−1, 0) → 180.00°, not 0°.

  • Precision

    r to four decimals; theta to two.

Frequently asked questions

  • What does this rectangular to polar calculator find?

    Polar coordinates (r, θ) from rectangular coordinates (x, y). r = √(x² + y²) and θ = atan2(y, x) converted to degrees. For x=3, y=4 you get r = 5.0000 and theta = 53.13°. r uses four decimal places; theta uses two.

  • What is r?

    The distance from the origin to the point. It is always at least 0. The same formula as a rectangle diagonal or vector magnitude: √(x² + y²). x=5, y=12 gives r = 13.0000.

  • What is theta?

    The angle from the positive x-axis to the ray through (x, y), in degrees. The page uses atan2(y, x), which places the angle in the correct quadrant. x=1, y=1 gives 45.00°. x=0, y=1 gives 90.00°.

  • Why use atan2 instead of atan(y/x)?

    atan(y/x) loses quadrant information when x is negative or zero. atan2(y, x) keeps the signs of both coordinates. x=−1, y=0 gives 180.00°, not 0°. x=0, y=−1 gives −90.00°.

  • What angle range does theta use?

    Degrees from atan2, typically from −180° to 180°. Points below the x-axis can show a negative angle, such as −90.00° for (0, −1). Points on the negative x-axis show 180.00°.

  • What about the origin (0, 0)?

    r is 0.0000. atan2(0, 0) is treated as 0 on this page, so theta shows 0.00°. The direction of a zero-length vector is undefined in geometry; the calculator still prints those values.

  • What inputs are required?

    Both x and y must be filled with finite numbers. Negatives and decimals are allowed. An empty field leaves both r and theta as “--.”

  • What is (−3, 4) in polar form?

    r = √(9 + 16) = 5.0000. The point is in the second quadrant, so theta = 126.87°. That matches atan2(4, −3) in degrees.

  • How do I go back to rectangular coordinates?

    x = r cos(θ) and y = r sin(θ) with θ in the same degree measure converted to radians for the trig functions. For r=5 and θ≈53.13°, you recover about x=3 and y=4.

  • Is theta in radians?

    No. The display converts atan2’s radian result to degrees with × 180 ÷ π and labels it with °. 90.00° is a right angle above the positive x-axis.