terican

BIPM-ratified constants · v1.0

Converter

Polar, form calculator.

Convert between rectangular (x, y) and polar (r, θ) coordinates. Supports degrees and radians with instant, accurate results.

From

magnitude r

magnitude

3 magnitude =5Computed Value

Equivalents

Precision: 6 dp · Notation: Decimal · 2 units
Degreesdegrees5
Radiansradians5

Common pairings

1 magnitudeequals4.1231 degrees
1 magnitudeequals4.1231 radians
1 angleequals75.9638 degrees
1 angleequals1.3258 radians
1 x_componentequals3 degrees
1 x_componentequals-4.8094 radians
1 y_componentequals4 degrees
1 y_componentequals1.3672 radians

The conversion

How the value
is computed.

Polar Form Calculator: Formulas, Variables, and Applications

Polar coordinates provide an alternative two-dimensional coordinate system that is often more natural than rectangular (Cartesian) coordinates for problems involving circles, spirals, rotation, and radial symmetry. Every point in the plane maps to a unique pair (r, θ) when r ≥ 0 and −π < θ ≤ π. This polar form converter handles both conversion directions — rectangular to polar and polar to rectangular — using the standard trigonometric relationships.

Core Conversion Formulas

Four fundamental equations connect the rectangular coordinates (x, y) and the polar coordinates (r, θ). These follow directly from the definition of sine, cosine, and the Pythagorean theorem applied to the right triangle formed by x, y, and r:

  • Magnitude (rectangular → polar): r = √(x² + y²)
  • Angle (rectangular → polar): θ = atan2(y, x)
  • Horizontal component (polar → rectangular): x = r · cos(θ)
  • Vertical component (polar → rectangular): y = r · sin(θ)

The magnitude formula is simply the Euclidean distance from the origin to the point (x, y). The angle formula uses the two-argument arctangent atan2(y, x), which resolves all four quadrants correctly — unlike arctan(y/x), which returns values only in (−90°, 90°) and loses quadrant information.

Variable Definitions

  • x — The horizontal Cartesian coordinate. Points to the right of the origin are positive; points to the left are negative.
  • y — The vertical Cartesian coordinate. Points above the origin are positive; points below are negative.
  • r (magnitude) — The radial distance from the origin to the point. Always non-negative (r ≥ 0) under the standard convention.
  • θ (theta) — The polar angle measured counterclockwise from the positive x-axis to the ray pointing at the given point. Expressed in degrees or radians.

Worked Example: Rectangular to Polar

Given the point (x, y) = (3, 4):

  • r = √(3² + 4²) = √(9 + 16) = √25 = 5
  • θ = atan2(4, 3) ≈ 53.13° (≈ 0.9273 radians)

The polar form is (r, θ) = (5, 53.13°). This matches the classic 3-4-5 Pythagorean triple, providing a useful geometric check.

Worked Example: Polar to Rectangular

Given (r, θ) = (10, 30°):

  • x = 10 · cos(30°) = 10 · (√3 / 2) ≈ 8.6603
  • y = 10 · sin(30°) = 10 · (1/2) = 5.0000

The rectangular form is approximately (8.6603, 5.0000), verifiable by computing √(8.6603² + 5²) = √(75 + 25) = √100 = 10.

Quadrant Awareness and the atan2 Function

A critical distinction: atan2(y, x) returns an angle in the full range (−180°, 180°], while arctan(y/x) only covers (−90°, 90°). The point (−3, −4) has θ = atan2(−4, −3) ≈ −126.87°, correctly placing it in the third quadrant. Using arctan alone would yield arctan(4/3) ≈ 53.13° — an entirely wrong quadrant assignment. As explained in Paul's Online Math Notes — Calculus II: Polar Coordinates, careful quadrant handling is essential when converting from rectangular to polar form.

Real-World Applications

  • Electrical Engineering: AC circuit analysis expresses impedances as phasors in polar form Z = r∠θ. An impedance of Z = 3 + 4j Ω converts to Z = 5∠53.13° Ω, making series and parallel calculations significantly simpler.
  • Robotics and Navigation: Radar and sonar systems report target positions as (range, bearing) — a natural polar format that requires rectangular conversion for grid-based mapping, path planning, and sensor fusion.
  • Complex Numbers: A complex number z = x + iy has polar form z = r · e^(iθ) per Euler's formula. Multiplying two complex numbers reduces to multiplying magnitudes and adding angles, far simpler than rectangular expansion.
  • Physics and Engineering Design: Central force problems (gravity, electrostatics) and wave propagation equations simplify substantially in polar or cylindrical coordinates, where radial symmetry matches the underlying physics.

Angle Units: Degrees vs. Radians

Radians are the natural unit for calculus and most physics formulas. Degrees are standard in engineering drawings, navigation, and everyday geometry. One full revolution equals 2π ≈ 6.2832 radians = 360°. To convert an atan2 result in radians to degrees, multiply by 180/π ≈ 57.2958. This calculator supports both units so results integrate directly into downstream work without manual conversion.

Methodology and Sources

This polar form converter implements the standard trigonometric conversion formulas documented in Paul's Online Math Notes — Calculus II: Polar Coordinates and MFG Polar Coordinates (University of Nebraska–Lincoln). The atan2 function follows IEEE 754 conventions, returning values in (−π, π] radians and correctly handling boundary cases such as x = 0 (yielding ±90°) and the origin (r = 0, θ = 0).

Reference

Frequently asked questions

What is polar form and how does it differ from rectangular form?
Polar form expresses a point using its distance from the origin (r) and its angle from the positive x-axis (θ). Rectangular form uses horizontal and vertical coordinates (x, y). Polar form simplifies complex number multiplication and is the natural representation for rotation-based problems in physics, robotics, and electrical engineering, where magnitude and direction matter more than grid position.
How do you convert rectangular coordinates to polar form?
To convert (x, y) to polar form, calculate the magnitude r = √(x² + y²) and the angle θ = atan2(y, x). For example, the point (3, 4) gives r = √(9 + 16) = 5 and θ = atan2(4, 3) ≈ 53.13°. Always use atan2 rather than plain arctan to correctly determine the quadrant of the resulting angle.
Why use atan2 instead of arctan for polar conversion?
The standard arctan function returns values only between −90° and 90°, so it cannot distinguish between opposite quadrants. For instance, both (1, 1) and (−1, −1) yield arctan(y/x) = 45°, but their true angles are 45° and −135° respectively. The two-argument atan2(y, x) considers the signs of both coordinates independently, returning the correct angle across all four quadrants without any ambiguity.
What does the angle θ represent in polar coordinates?
The angle θ is measured counterclockwise from the positive x-axis to the ray connecting the origin to the point. Using atan2, it ranges from −180° to 180° (or −π to π radians). In navigation, bearings are measured clockwise from north, so converting polar θ to a compass bearing requires subtracting θ from 90° and adjusting the result into a 0°–360° range.
How is polar form used in electrical engineering?
Electrical engineers represent AC voltages, currents, and impedances as phasors in polar form: Z = r∠θ, where r is the magnitude in ohms and θ is the phase angle in degrees. For example, the impedance Z = 3 + 4j Ω converts to Z = 5∠53.13° Ω. Multiplying impedances in polar form reduces to multiplying magnitudes and adding angles, which is far simpler than expanding rectangular arithmetic.
Can r be negative in polar coordinates?
By mathematical convention, r can be assigned a negative value, meaning the point lies in the direction opposite to θ. For example, (r, θ) = (−3, 45°) represents the same point as (3, 225°). However, the standard convention used in engineering and complex number analysis keeps r ≥ 0, producing a unique, unambiguous representation. This calculator always returns a non-negative magnitude.