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?

1 Answer
George C.
May 12, 2016

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.

color(white)()
"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...

enter image source here

Related questions
See all questions in Linear, Exponential, and Quadratic Models
Impact of this question
3461 views around the world
You can reuse this answer
Creative Commons License