BIPM-ratified constants · v1.0

Converter

Rectangular, to polar coordinate converter calculator.

Convert (x, y) rectangular coordinates to polar form (r, θ) using r = √(x²+y²) and θ = atan2(y, x). Supports both radians and degrees.

From

Y Coordinate

Angle Unit

Equivalents

Precision: 6 dp · Notation: Decimal · 2 units

Magnitude (r)r5

Angle (θ)theta53.1301

The conversion

How the value
is computed.

Understanding Rectangular to Polar Coordinate Conversion

Two fundamental systems exist for locating points in a two-dimensional plane: the rectangular (Cartesian) coordinate system, which uses a horizontal distance (x) and vertical distance (y) from an origin, and the polar coordinate system, which describes a point by its radial distance (r) from the origin and an angle (θ) measured counterclockwise from the positive x-axis. Converting between these systems is a foundational operation in precalculus, calculus, physics, and electrical engineering.

The Core Conversion Formulas

Given a point with rectangular coordinates (x, y), the polar coordinates (r, θ) are computed as follows:

Magnitude: r = √(x² + y²)
Angle: θ = atan2(y, x)

Deriving the Magnitude Formula

Drawing a point (x, y) on the Cartesian plane and connecting it to the origin creates a right triangle with legs of length |x| and |y|. By the Pythagorean theorem, the length of the hypotenuse — the straight-line distance from the origin to the point — equals √(x² + y²). For example, the point (3, 4) yields r = √(9 + 16) = √25 = 5 units.

Why atan2 Instead of arctan?

The angle θ is the direction from the origin to the point. A naive approach uses arctan(y/x), but this function returns values only in the range (−π/2, π/2), covering just Quadrants I and IV, and is completely undefined when x = 0. The two-argument function atan2(y, x) uses the individual signs of both y and x to determine the correct quadrant, returning a unique angle in (−π, π]. As noted by Pauls Online Math Notes — Calculus II: Polar Coordinates, proper quadrant determination is critical for an unambiguous conversion. This is why virtually all programming languages and scientific calculators implement atan2 rather than a simple arctan quotient.

Variables and Their Meaning

x (X Coordinate): Horizontal displacement from the origin. Positive x is to the right; negative x is to the left.
y (Y Coordinate): Vertical displacement from the origin. Positive y is upward; negative y is downward.
r (Radial Distance / Magnitude): The non-negative distance from the origin to the point. Always r ≥ 0. When x = 0 and y = 0, r = 0.
θ (Polar Angle / Argument): The angle swept counterclockwise from the positive x-axis to the radial line. Returned in (−π, π] radians or (−180°, 180°] degrees.

Radians vs. Degrees

The angle θ can be expressed in radians or degrees. Radians are the natural unit in mathematics and physics — a full revolution equals 2π ≈ 6.2832 rad — and must be used in calculus formulas such as arc length and area in polar form. Degrees offer more intuitive visualization for many users. To convert: multiply radians by 180/π to obtain degrees. For example, θ = π/4 rad = 45°, and θ = 2π/3 rad ≈ 120°.

Worked Examples

Example 1 — Quadrant I: Point (3, 4)

r = √(9 + 16) = 5. θ = atan2(4, 3) ≈ 0.9273 rad ≈ 53.13°. The point lies 5 units from the origin at 53.13° above the positive x-axis.

Example 2 — Quadrant II: Point (−5, 5)

r = √(25 + 25) = √50 ≈ 7.071. θ = atan2(5, −5) = 3π/4 rad = 135°. A simple arctan(5/−5) = arctan(−1) = −45° would place the point incorrectly in Quadrant IV; atan2 correctly identifies Quadrant II.

Example 3 — Negative y-axis: Point (0, −6)

r = 6. θ = atan2(−6, 0) = −π/2 rad = −90°. The arctan function is undefined when x = 0; atan2 handles this boundary case precisely.

Applications Across Disciplines

Polar coordinates appear throughout science and engineering. In electrical engineering, complex numbers representing voltage phasors and impedances are expressed as magnitude and phase angle — a direct application of this conversion, as explained in Gordy — Rectangular and Polar Complex Numbers (TCC). In robotics and autonomous navigation, lidar sensors return range-and-bearing (polar) readings that algorithms convert to Cartesian for mapping. In antenna design, radiation patterns are naturally plotted in polar form. In computer graphics, converting to polar facilitates radial gradient computation and spiral path generation. Mastering both coordinate systems and the conversion between them unlocks modeling power across all of these fields.

Reference

Frequently asked questions

What is the difference between rectangular and polar coordinates?

Rectangular (Cartesian) coordinates describe a point using its horizontal distance x and vertical distance y from an origin. Polar coordinates describe the same point using a radial distance r from the origin and an angle θ measured counterclockwise from the positive x-axis. Both systems represent any point in the plane, but polar form is more natural for circular, rotational, and wave-based problems. For example, the point (3, 4) in rectangular form becomes (5, 53.13°) in polar form.

Why does this calculator use atan2 instead of the simple arctan formula?

The basic arctan(y/x) function only returns angles between −90° and 90°, covering only Quadrants I and IV, and is undefined when x = 0. The atan2(y, x) function examines the signs of both y and x independently to place the angle in the correct quadrant, returning a result in the full range (−180°, 180°]. For the point (−5, 5) in Quadrant II, arctan yields −45° (incorrect), while atan2 correctly returns 135°. This makes atan2 the standard for robust, unambiguous coordinate conversion.

How do I convert the point (−3, −3) to polar coordinates step by step?

Step 1 — compute the magnitude: r = √((−3)² + (−3)²) = √(9 + 9) = √18 ≈ 4.243. Step 2 — compute the angle: θ = atan2(−3, −3) = −135° (equivalently −3π/4 ≈ −2.356 rad). The point lies in Quadrant III, approximately 4.243 units from the origin at an angle of −135° clockwise from the positive x-axis. The final polar form is (4.243, −135°) in degrees or (4.243, −2.356) in radians.

What range of values can the angle θ take in the output?

When computed with atan2, the angle θ falls in the half-open interval (−π, π] radians, corresponding to (−180°, 180°] in degrees. Angles for points in Quadrants I and II are positive (0° to 180°), while angles for points in Quadrants III and IV are negative (−180° to 0°). The exact value −π (or −180°) is excluded because that boundary is shared with π (180°) at the negative x-axis, ensuring every point maps to exactly one angle output.

When is it better to use polar coordinates instead of rectangular coordinates?

Polar coordinates simplify problems with circular or rotational symmetry. In calculus, computing the area enclosed by a rose curve or limaçon is far easier in polar form than rectangular form. In physics, circular motion, gravitational potential wells, and electromagnetic fields around a straight wire are naturally described using radius and angle. In electrical engineering, phasor analysis of AC circuits uses magnitude and phase angle — the polar representation of complex impedance. Whenever distance from a central point and angular direction matter more than horizontal and vertical offsets, polar coordinates are the preferred choice.

Can the radial distance r be negative, and what does the origin (0, 0) produce?

In the standard mathematical convention this calculator follows, the radial distance r is always non-negative (r ≥ 0) because it represents a Euclidean distance computed via the Pythagorean theorem. Some advanced treatments permit a negative r by reflecting the point 180° through the origin, but that convention is non-standard. For the special case of the origin (0, 0), the magnitude r equals 0, and the angle θ is mathematically undefined — atan2(0, 0) has no standard value because the origin is the single point in the plane with no well-defined direction.