Question

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?

Answer 1

`50` square inches

Explanation:

If a circle has radius `r` then:

• Its circumference is `2pi r`

• Its area is `pi r^2`

An arc of length `r` is `1/(2pi)` of the circumference.

So the area of a sector formed by such an arc and two radii will be `1/(2pi)` multiplied by the area of the whole circle:

`1/(2pi) xx 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.

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"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 `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 `2pi r`) is `pi r^2`...