Polar Coordinates & Area

Graph r = f(θ) and find enclosed area with 12∫ r^2 dθ.

The idea

Polar coordinates locate a point in the plane by its distance $r$ from the origin $O$ and the angle $\theta$ from the positive $x$-axis to the ray through the point. Drawing the right triangle from the point down to the $x$-axis converts between polar and Cartesian: $x = r\cos\theta, \qquad y = r\sin\theta, \qquad r^{2} = x^{2} + y^{2}.$ An equation $r = f(\theta)$ describes a polar curve, traced as $\theta$ sweeps around the origin. Curves organized around a center — circles through the origin, spirals, roses — have short polar equations and awkward Cartesian ones.

To find the area of a region swept out from $\theta = \alpha$ to $\theta = \beta$, cut it into thin wedges. A wedge of angle $d\theta$ reaching out to radius $r$ is very nearly a circular sector: the fraction $\frac{d\theta}{2\pi}$ of a disc of area $\pi r^{2}$, which is $\frac{1}{2} r^{2}\,d\theta$. Adding the wedges gives $A = \frac{1}{2}\int_{\alpha}^{\beta} r^{2}\,d\theta.$ The integrand carries $r^{2}$, not $r$: a wedge twice as long has four times the area.

Ways to work on it

• Walkthrough. Convert points, read a polar circle, and apply the area formula.
• Practice. Area enclosed by a polar circle r = a θ or r = a θ.
• Hardest. Area of one petal of a rose via the polar area integral.

Not sure where to start? Take the ten-question placement test.