Shown is a sector of a circle. If its perimeter is 39 millimeters, what is its area?
There are two formulas we need to recall in order to be able to solve this problem. The first is the formula for the arc length of a sector with an angle of 𝜃 radians. Arc length equals 𝑟 times 𝜃. The second is the formula for the area of this sector. Sector area equals a half times 𝑟 squared times 𝜃.
Now, we are given the perimeter of the shape and asked to find the area. In order to do this, first, we will need to establish the size of the angle 𝜃. 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. If we subtract the value of the two radii, that will leave us with just the arc length. Arc length is 39 minus nine add nine, which is 21 millimeters.
Let’s now substitute everything we know into our formula for arc length. The arc length is 21 millimeters and the radius is nine millimeters. So 21 equals nine times 𝜃. Dividing through by nine gives us 21 over nine or seven over three radians.
The question actually wants us to find the area of the sector. We know the radius is nine millimeters and the angle is seven over three radians. Our formula for the area becomes a half times nine squared times seven over three.
The area of our sector is 94.5 millimeters squared.