WEBVTT
00:00:02.640 --> 00:00:04.620
Shown is a sector of a circle.
00:00:04.960 --> 00:00:08.940
If its perimeter is 39 millimeters, what is its area?
00:00:10.590 --> 00:00:14.980
There are two formulas we need to recall in order to be able to solve this problem.
00:00:16.150 --> 00:00:21.610
The first is the formula for the arc length of a sector with an angle of π radians.
00:00:22.150 --> 00:00:24.610
Arc length equals π times π.
00:00:25.990 --> 00:00:29.110
The second is the formula for the area of this sector.
00:00:29.540 --> 00:00:33.920
Sector area equals a half times π squared times π.
00:00:35.770 --> 00:00:39.610
Now, we are given the perimeter of the shape and asked to find the area.
00:00:40.440 --> 00:00:45.500
In order to do this, first, we will need to establish the size of the angle π.
00:00:47.450 --> 00:00:57.490
We also have to remember that perimeter is slightly different to the arc length in that it is the distance around the entire shape, whereas the arc length is just the curved part.
00:00:59.110 --> 00:01:03.910
If we subtract the value of the two radii, that will leave us with just the arc length.
00:01:04.760 --> 00:01:09.530
Arc length is 39 minus nine add nine, which is 21 millimeters.
00:01:11.760 --> 00:01:15.770
Letβs now substitute everything we know into our formula for arc length.
00:01:17.220 --> 00:01:21.590
The arc length is 21 millimeters and the radius is nine millimeters.
00:01:21.740 --> 00:01:24.460
So 21 equals nine times π.
00:01:26.140 --> 00:01:31.420
Dividing through by nine gives us 21 over nine or seven over three radians.
00:01:33.370 --> 00:01:36.880
The question actually wants us to find the area of the sector.
00:01:37.470 --> 00:01:42.290
We know the radius is nine millimeters and the angle is seven over three radians.
00:01:43.790 --> 00:01:49.300
Our formula for the area becomes a half times nine squared times seven over three.
00:01:51.040 --> 00:01:55.740
The area of our sector is 94.5 millimeters squared.