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Polar Plot — Definition, Formula & Examples

A polar plot is a graph drawn on a polar coordinate system, where each point is determined by its distance rr from the origin and its angle θ\theta from the positive xx-axis.

A polar plot is the set of all points (r,θ)(r, \theta) satisfying a given polar equation r=f(θ)r = f(\theta), rendered on a plane with concentric circles marking radial distance and rays from the origin marking angular position.

How It Works

To create a polar plot, you choose a set of θ\theta values (typically from 00 to 2π2\pi), compute r=f(θ)r = f(\theta) for each, and then place each point at distance rr from the origin along the direction θ\theta. Positive rr means the point lies along the ray at angle θ\theta; negative rr means it lies along the opposite ray. You connect the resulting points smoothly to reveal the curve. Symmetry tests (replacing θ\theta with −θ-\theta, or rr with −r-r) can reduce the work by identifying mirror symmetry about axes or the origin.

Worked Example

Problem: Plot the polar equation r=2cos⁡θr = 2\cos\theta by evaluating several key angles.
Evaluate key points: Compute rr at selected angles:
θ=0:  r=2cos⁡0=2θ=π3:  r=2cos⁡π3=1θ=π2:  r=2cos⁡π2=0\theta = 0:\; r = 2\cos 0 = 2 \qquad \theta = \frac{\pi}{3}:\; r = 2\cos\frac{\pi}{3} = 1 \qquad \theta = \frac{\pi}{2}:\; r = 2\cos\frac{\pi}{2} = 0
Note symmetry: Since cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta, the curve is symmetric about the polar axis (the positive xx-axis). Points for negative θ\theta mirror those for positive θ\theta.
Identify the shape: Converting to Cartesian: multiply both sides by rr to get r2=2rcos⁡θr^2 = 2r\cos\theta, which becomes x2+y2=2xx^2 + y^2 = 2x. Completing the square gives a circle.
(x−1)2+y2=1(x-1)^2 + y^2 = 1
Answer: The polar plot of r=2cos⁡θr = 2\cos\theta is a circle of radius 1 centered at (1,0)(1, 0).

Why It Matters

Polar plots are essential for graphing curves that have natural rotational or radial symmetry, such as cardioids, roses, and lemniscates. In physics and engineering, polar plots represent antenna radiation patterns, wind direction data, and orbital paths far more naturally than Cartesian graphs.

Common Mistakes

Mistake: Ignoring negative rr values and skipping those points entirely.
Correction: When r<0r < 0, plot the point at distance ∣r∣|r| in the direction opposite to θ\theta. This produces valid parts of the curve, such as inner loops on a limaçon.