Polar Coordinates Formulas
Convert between polar and cartesian coordinates using r, theta, x, and y.
Learn polar coordinate formulas with worked examples.
Polar to Cartesian
y = r sin θ
To convert from polar coordinates (r, θ) to cartesian coordinates (x, y), use the cosine and sine functions. The radius r is the distance from the origin, and θ is the angle measured counterclockwise from the positive x-axis.
Cartesian to Polar
θ = arctan(y / x)
To convert from cartesian to polar, find the distance using the Pythagorean theorem and the angle using the arctangent. Be careful with the arctangent — you may need to adjust the angle depending on which quadrant the point is in.
For points not in the first quadrant, use the atan2 function (available in most calculators and programming languages) which automatically handles all four quadrants.
Variables
| Symbol | Meaning |
|---|---|
| r | Radial distance from the origin (always non-negative) |
| θ | Angle measured counterclockwise from the positive x-axis (in radians or degrees) |
| x | Horizontal coordinate in cartesian system |
| y | Vertical coordinate in cartesian system |
Quadrant Adjustments for θ
- Quadrant I (x > 0, y > 0): θ = arctan(y/x)
- Quadrant II (x < 0, y > 0): θ = arctan(y/x) + π
- Quadrant III (x < 0, y < 0): θ = arctan(y/x) + π
- Quadrant IV (x > 0, y < 0): θ = arctan(y/x) + 2π
Example 1: Polar to Cartesian
Convert the polar point (5, 60°) to cartesian coordinates.
Identify values: r = 5, θ = 60°
x = r cos θ = 5 × cos 60° = 5 × 0.5 = 2.5
y = r sin θ = 5 × sin 60° = 5 × 0.866 = 4.33
(x, y) = (2.5, 4.33)
Example 2: Cartesian to Polar
Convert the cartesian point (−3, 4) to polar coordinates.
Calculate r: r = √(x² + y²) = √((−3)² + 4²) = √(9 + 16) = √25 = 5
Calculate θ: arctan(y/x) = arctan(4/(−3)) = arctan(−1.333) = −53.13°
The point is in Quadrant II (x < 0, y > 0), so add 180°: θ = −53.13° + 180° = 126.87°
(r, θ) = (5, 126.87°) or approximately (5, 2.214 radians)
When to Use These
Polar coordinates are useful when problems have circular or rotational symmetry.
- Describing circular motion and orbits
- Working with radar and sonar systems (distance and bearing)
- Graphing roses, spirals, cardioids, and other polar curves
- Simplifying integrals with circular boundaries
- Navigation using distance and heading