Area and Arc Length in Polar Coordinates

Arc Length

If you were to straighten a curved line out, the measured length would be the arc length. Since it can be very difficult to measure the length of an arc linearly, the solution is to use polar coordinates. Using polar coordinates allows us to integrate along the length of the arc in order to compute its length.

The arc length of the curve defined by a polar function is found by the integration over the curve $r(\theta)$. Let $L$ denote this length along the curve starting from points $A$ through to point $B$, where these points correspond to $\theta = a$ and $\theta = b$ such that $0 < b-a < 2 \pi$. The length of $L$ is given by the following integral:

$\displaystyle{L = \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta}$

Solving this integral will give the length of the arc.

Arc Length

The curved lines bounding the region $R$ are arcs. Their length can be calculated with calculus. The area of the region $R$ can also be calculated by integration.

Arc Segment Area

To find the area enclosed by the arcs and the radius and polar angles, you again use integration.

Let $R$ denote the region enclosed by a curve $r(\theta)$ and the rays $\theta = a$ and $\theta = b$, where $0 < b-a < 2 \pi$. Then, the area of $R$ is:

$\displaystyle{\frac{1}{2} \int_a^b r^2 d\theta}$

This result can be found as follows. 

First, the interval $[a, b]$ is divided into $n$ subintervals, where $n$ is an arbitrary positive integer. Thus $\Delta \theta$, the length of each subinterval, is equal to $b-a$ (the total length of the interval), divided by $n$, the number of subintervals. For each subinterval $i = 1, 2, \cdots , n$, let $\theta_i$ be the midpoint of the subinterval, and construct a sector with the center at the pole, radius $r(\theta_i)$, central angle $\Delta \theta$ and arc length $r(\theta_i)\Delta\theta$. The area of each constructed sector is therefore equal to

$\displaystyle{\frac{1}{2} r^2 \Delta\theta}$

And the total area is the sum of these sectors. An infinite sum of these sectors is the same as integration.