What is the area of a sector of a circle that has a diameter of 10 in. if the length of the arc is 10 in?

May 12, 2016

$50$ square inches

Explanation:

If a circle has radius $r$ then:

• Its circumference is $2 \pi r$

• Its area is $\pi {r}^{2}$

An arc of length $r$ is $\frac{1}{2 \pi}$ of the circumference.

So the area of a sector formed by such an arc and two radii will be $\frac{1}{2 \pi}$ multiplied by the area of the whole circle:

$\frac{1}{2 \pi} \times \pi {r}^{2} = {r}^{2} / 2$

In our example, the area of the sector is:

(10"in")^2/2 = (100"in"^2) / 2 = 50"in"^2

$50$ square inches.

$\textcolor{w h i t e}{}$
"Paper and Scissors" Method

Given such a sector, you could cut it up into an even number of sectors of equal size, then rearrange them head to tail to form a slightly "bumpy" parallelogram. The more sectors you cut it into, the closer the parallelogram would be to a rectangle with sides $r$ and $\frac{r}{2}$ and thus area ${r}^{2} / 2$.

I don't have a picture for that, but here's an animation I put together that shows a similar process with a whole circle, illustrating that the area of a circle (which has circumference $2 \pi r$) is $\pi {r}^{2}$... 