Rectangular to Polar Calculator - (x, y) to (r, theta) Converter

A rectangular to polar calculator turns a point written as (x, y) on the rectangular coordinate plane into the same point written as (r, theta), where r is the distance from the origin and theta is the angle measured counterclockwise from the positive x-axis. It applies the two formulas r = sqrt(x^2 + y^2) and theta = atan2(y, x), so you can read both the radius and the angle in degrees and radians from any input, plus a quadrant label to confirm which side of the origin the point is on. Use it whenever a problem gives you rectangular (x, y) coordinates and expects a polar answer, or whenever you need a clean r and theta for a force vector, a phasor, or a circular-motion diagram.

• • Convert a rectangular point to polar form: A point written as (3, 4) becomes r = 5, theta = 53.13 degrees in a single step, the familiar 3-4-5 right triangle expressed in polar form.
• • Read the magnitude and direction of a 2D vector: Given the components of a force or velocity vector, the calculator returns the vector length r and the heading theta. The matching Vector Magnitude Calculator covers the same r = sqrt(x^2 + y^2) length when you only need the size.
• • Match the language of an older textbook or calculus problem: Some textbooks call the (x, y) plane the rectangular plane rather than the Cartesian plane. The calculator uses the same formulas, so the output matches the cartesian-to-polar version exactly.

The rectangular coordinate plane labels every point with a horizontal x and a vertical y, while the polar system labels the same point with a distance from the origin and an angle from a reference direction. They describe the same plane, just with a different set of rulers, and the conversion is exact rather than approximate.

The term 'rectangular' is the older, geometry-style name for the (x, y) coordinate system, and 'Cartesian' is the more common name in modern algebra and calculus. Either spelling is correct for the same plane.

Because the rectangular plane and the Cartesian plane are the same (x, y) plane under a different name, the Cartesian to Polar Calculator on this site returns the identical r and theta for any (x, y) input you enter here.

The calculator takes x and y, builds the radial distance r with the Pythagorean theorem, and recovers the angle with the two-argument arctangent atan2(y, x). It then multiplies the radian result by 180/pi to get the degree value, and picks a quadrant label from the signs of x and y. The results appear side by side so the polar answer can be read at a glance and pasted straight into a degree-based or radian-based formula.

• x: Horizontal rectangular coordinate. Positive values are to the right of the y-axis, negative values to the left.
• y: Vertical rectangular coordinate. Positive values are above the x-axis, negative values below.
• r (radius): Distance from the origin to (x, y). Always non-negative; computed as the Pythagorean length of x and y.
• theta (angle): Angle measured counterclockwise from the positive x-axis. In degrees the result lives in (-180°, 180°]; in radians it lives in (-pi, pi].

Math.atan2 is the standard two-argument inverse tangent built into JavaScript, Python, spreadsheets, and most scientific libraries. Its return value lives in the open interval (-pi, pi] (about -180° to 180°), and the four sign combinations of (x, y) place the angle in quadrant I, II, III, or IV, which is exactly the convention the calculator surfaces in the Quadrant row.

According to Wolfram MathWorld, the polar coordinates of a rectangular point (x, y) are (r, theta) where r = sqrt(x^2 + y^2) is the radial distance from the origin and theta = atan2(y, x) is the angle measured counterclockwise from the positive x-axis

According to Wikipedia (Polar coordinate system), the polar coordinate system represents a point by its distance r from a pole (the origin) and the angle theta from a polar axis (the positive x-axis), with the rectangular point recovered from x = r cos(theta) and y = r sin(theta)

When you only need the angle from a y/x ratio rather than a full (x, y) point, the Arcus Tangent Calculator applies the same atan2-based quadrant logic on its arctan2 tab.

Four ideas cover every rectangular-to-polar conversion you will run into, from a homework problem to a phasor diagram.

The radial distance and the angle are two halves of the same point. Knowing r and theta is enough to recover x = r cos(theta) and y = r sin(theta); knowing x and y is enough to recover r and theta through the formulas on this page.

Because r is just the hypotenuse of the right triangle formed by x and y, Right Triangle Calculator is the natural place to solve the matching legs and angles for the same input.

Five short steps take any rectangular (x, y) input to its polar (r, theta) answer, including the trickier cases on the axes and at the origin.

1. 1 Enter x and y: Type the horizontal and vertical coordinates. Positive, negative, zero, and decimals all work; any real pair is accepted.
2. 2 Read the radius: The top result is r, the distance from the origin to (x, y). It is always a non-negative number, regardless of the signs of x and y.
3. 3 Read the angle in degrees: The second result is theta in degrees, the angle from the positive x-axis to the point, in the range (-180°, 180°].
4. 4 Read the angle in radians: The third result is the same theta in radians, which is the unit every programming language and scientific library expects.
5. 5 Check the quadrant label: The bottom result says I, II, III, IV, the axis label (+x, -x, +y, -y), or 'Origin' when both inputs are zero. It is the fastest way to confirm the answer is on the side of the plane you expected.

A drone flies 80 meters east and 30 meters north from a field base. Type x = 80 and y = 30 into the calculator. The radius reads 85.44 meters and theta reads 20.56 degrees, the heading measured counterclockwise from due east.

The rectangular to polar calculator handles the four operations you would do by hand, plus the sign and axis cases that usually trip people up.

• • Both angle units in one step: Degrees and radians appear in the results panel at the same time, so there is no need to remember the 180/pi factor mid-calculation.
• • Quadrant label built in: The calculator returns a roman-numeral I, II, III, or IV, or an axis label, so you can confirm the side of the plane without drawing a coordinate diagram.
• • Atan2 used automatically: The two-argument arctangent picks the correct angle for every (x, y) input, including the four quadrants and the two axes where the single-argument arctan would silently return 0.
• • Domain error handled at the origin: When both x and y are zero, the calculator returns r = 0 and an explicit 'Undefined' angle rather than a silent NaN, so the origin is never a confusing edge case.
• • Pairs with related coordinate tools: The same r and theta feed directly into the right triangle calculator for side solving, the complex number calculator for argument, and the angle converter for gradians or DMS.

If the (x, y) pair is the real and imaginary parts of a complex number, Complex Number Calculator converts the same input to polar form including the argument in degrees and radians.

Three things change the answer you should expect, plus two practical caveats about how the rectangular-to-polar conversion is usually defined.

• • The polar form of a non-origin point is not unique: (r, theta) and (r, theta + 360 degrees) describe the same point. The calculator reports the principal value in (-180°, 180°], so add or subtract full turns yourself if your problem expects a positive-only range.
• • The angle is rounded to 4 decimal places in degrees and 6 in radians. That is more than enough for engineering and academic work, but high-precision metrology may want to extend the displayed precision or use a dedicated tool.

When the radius r is exactly zero, the angle is mathematically undefined and the calculator says so in plain text. Treat the origin as a special case rather than a placeholder for a zero angle.

According to Wikipedia (Atan2), atan2(y, x) returns the angle whose tangent is y/x and whose sign follows the signs of y and x, with the result in the open interval (-pi, pi] so that every (x, y) pair maps to a unique theta

If you need the angle in gradians or DMS instead of the two units already shown, Radians to Degrees Calculator is the next step in the chain without redoing the rectangular-to-polar math.