Cartesian to Polar Calculator
Convert between Cartesian (x,y) and Polar (r,theta) coordinate systems.
Conversion Mode
Coordinate Plane
Polar Coordinates
(5.0000, 53.13)
Quadrant 1
Conversion Formulas
Cartesian to Polar
r = sqrt(x^2 + y^2)
theta = atan2(y, x)
Polar to Cartesian
x = r x cos(theta)
y = r x sin(theta)
Understanding Coordinate Systems
Cartesian Coordinates
Uses perpendicular x and y axes to specify a point's location by its horizontal and vertical distances from the origin.
Polar Coordinates
Uses distance from origin (r) and angle from positive x-axis (theta) to specify a point's location.
What Is a Cartesian to Polar Converter?
A Cartesian-to-polar converter transforms coordinates between the rectangular (x, y) system and the polar (r, θ) system. In Cartesian coordinates, a point is located by horizontal and vertical distances from the origin. In polar coordinates, the same point is described by its distance r from the origin and the angle θ from the positive x-axis. This calculator converts in both directions and supports both degrees and radians.
The conversion formulas are straightforward: r = √(x² + y²) and θ = atan2(y, x) for Cartesian to polar; x = r cos(θ) and y = r sin(θ) for polar to Cartesian. The calculator handles all quadrants correctly using the atan2 function and normalizes angles to the 0–360° (or 0–2π) range.
Conversion Formulas
Cartesian ↔ Polar Conversion
Where:
- x, y= Cartesian coordinates — horizontal and vertical distances from the origin
- r= Radial distance — the straight-line distance from the origin to the point
- θ= Angle from the positive x-axis — measured counterclockwise
How to Use
- Choose direction: Select Cartesian to Polar or Polar to Cartesian.
- Enter coordinates: For Cartesian, enter x and y. For polar, enter r and θ.
- Select angle unit: Degrees or radians for the θ value.
- Read both forms: The calculator always shows both representations so you can verify the conversion.
Applications
Polar coordinates simplify problems with radial symmetry — circles, spirals, and rotational motion are easier to describe in polar form. In physics, orbital mechanics and electromagnetic fields use polar/spherical coordinates. In engineering, antenna radiation patterns and cam profiles use polar coordinates. In computer graphics, circular layouts and radar displays use polar-to-Cartesian conversion.
Worked Examples
Cartesian to Polar
Problem:
Convert (3, 4) to polar coordinates.
Solution Steps:
- 1r = √(3² + 4²) = √(9+16) = √25 = 5.
- 2θ = atan2(4, 3) ≈ 53.13° (in degrees).
Result:
Polar: r = 5, θ ≈ 53.13° (or 0.9273 radians). This is a 3-4-5 triangle.
Polar to Cartesian
Problem:
Convert (r=5, θ=60°) to Cartesian.
Solution Steps:
- 1x = 5 × cos(60°) = 5 × 0.5 = 2.5.
- 2y = 5 × sin(60°) = 5 × 0.866 = 4.330.
Result:
Cartesian: (2.5, 4.330). The point lies in the first quadrant.
Negative r Handling
Problem:
Convert (r=−3, θ=45°) to Cartesian.
Solution Steps:
- 1Negative r means go in the opposite direction: effectively θ+180° at |r|.
- 2x = −3 × cos(45°) = −3 × 0.7071 = −2.121.
- 3y = −3 × sin(45°) = −3 × 0.7071 = −2.121.
Result:
(−2.121, −2.121), which is equivalent to (r=3, θ=225°).
Tips & Best Practices
- ✓(x, y) = (r·cos θ, r·sin θ) — memorize this pair; it's used everywhere from calculus to computer graphics.
- ✓atan2(y, x) is preferred over arctan(y/x) because it handles all quadrants and avoids division by zero at x=0.
- ✓Points on the x-axis have θ = 0° (positive x) or 180° (negative x); on the y-axis: 90° and 270°.
- ✓For points exactly at the origin (0, 0), r = 0 and θ is undefined — the calculator may return θ = 0.
Frequently Asked Questions
Sources & References
Last updated: 2026-06-06
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
Formula Source: Handbook of Mathematical Functions
by Abramowitz & Stegun