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

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