WEBVTT
00:00:01.760 --> 00:00:08.720
The radius of a circle is 40 centimeters and the arc length of a segment is 18 centimeters.
00:00:09.320 --> 00:00:12.720
Find the area of the segment giving the answer to two decimal places.
00:00:14.040 --> 00:00:16.320
Hereโs a diagram of the sector of our circle.
00:00:16.840 --> 00:00:24.600
We know its radius is 40 centimeters and its arc length is 18 centimeters.
00:00:25.480 --> 00:00:29.040
Now, weโre trying to find the area of the segment.
00:00:29.520 --> 00:00:31.280
Thatโs this bit shaded.
00:00:32.400 --> 00:00:40.320
And so, we recall that, to find the area of the segment, we find the area of the whole sector and then subtract the area of this triangle.
00:00:41.360 --> 00:00:47.520
The problem is, to find the area of both the sector and the triangle, we need to calculate the size of the angle ๐.
00:00:48.240 --> 00:00:56.640
And so to do so, weโre going to begin by using the information that the arc length of the segment is 18 centimeters.
00:00:57.760 --> 00:01:09.400
Now, if weโre working with degrees, the formula we use to find the arc length is ๐ times diameter multiplied by ๐ over 360.
00:01:10.960 --> 00:01:15.200
Essentially, we find a proportion of the full circumference of the circle.
00:01:16.160 --> 00:01:20.160
When weโre working with radians, though, things are much simpler.
00:01:20.960 --> 00:01:23.760
The formula we use is ๐๐.
00:01:24.600 --> 00:01:28.280
We multiply the length of the radius by the angle ๐ in radians.
00:01:28.880 --> 00:01:37.400
We know that the arc length of our segment is 18 centimeters and the radius of the circle is 40.
00:01:38.280 --> 00:01:44.800
So, substituting what we know into our formula, and we get 18 equals 40 times ๐.
00:01:45.600 --> 00:01:47.680
And then we divide both sides by 40.
00:01:48.400 --> 00:01:54.080
So, ๐ is 18 over 40, or nine twentieths, radians.
00:01:55.680 --> 00:01:59.760
And now we know the angle ๐ in radians.
00:02:00.560 --> 00:02:02.880
Weโre ready to calculate the area of the sector.
00:02:03.840 --> 00:02:08.400
When ๐ is measured in radians, the area is a half ๐ squared ๐.
00:02:08.880 --> 00:02:15.880
In this case then, the area of our sector is a half times 40 squared times nine twentieths.
00:02:16.600 --> 00:02:19.680
Thatโs 360 square centimeters.
00:02:20.520 --> 00:02:21.880
But what about the area of the triangle?
00:02:23.680 --> 00:02:33.680
Well, weโre going to use the trigonometric formula a half ๐๐ sin ๐, where ๐ is the vertex that sits at the center of our circle.
00:02:34.840 --> 00:02:40.440
The area of our triangle is a half times 40 times 40 times sin of nine twentieths.
00:02:41.360 --> 00:02:51.080
Thatโs 347.972, and weโre still working, of course, in square centimeters.
00:02:52.160 --> 00:02:57.440
We know that the area of the segment is the area of the sector minus the area of the triangle.
00:02:58.280 --> 00:03:07.080
So, thatโs 360 minus 347.972.
00:03:07.840 --> 00:03:13.240
Thatโs 12.0275.
00:03:14.000 --> 00:03:20.400
Correct to two decimal places, that rounds to 12.03.
00:03:21.080 --> 00:03:28.280
And we see then that the area of the segment of our circle is 12.03 square centimeters.