A circle with radius 27 centimetres has a sector cut from it. The perimeter of the sector is 102 centimetres. What is the area of the sector?
So to help us visualize what’s going on, I’ve drawn a circle with the sector cuts out. I’ve also shown that the radius is 27 centimetres. And we’re also told that the perimeter of the sector is 102 centimetres. Well, let’s think about the perimeter of our sector. The perimeter of our sector is gonna be the two radius lengths that I’ve marked on, added to the arc length. And I’ve called that arc length 𝑥. So therefore, I could say that the perimeter of this sector is equal to two 𝑟, cause that’s two multiplied by the radius, plus 𝑥, which is our arc length.
As 𝑥 is the only unknown, what I’m going to do is rearrange or change the subject of our formula to make 𝑥 the subject. So to do this, all I need to do is subtract two 𝑟 from each side of the formula. And when I do that, I get the perimeter minus two 𝑟 is gonna be equal to 𝑥.
Well, now what we can do is think about a formula we’ve got for arc length. Well, the formula is that the arc length is equal to 𝑟𝜃. But we must remember this formula only works when 𝜃 is in radians. So that’s the measurement of our angle. Okay, so if we’ve got the arc length equal to 𝑟𝜃, and we also know for our sector to the arc length 𝑥 is equal to 𝑃 minus two 𝑟, then we can say that the perimeter minus two 𝑟 must be equal to 𝑟𝜃 because, as we said, 𝑥 was the arc length. And the arc length is equal to 𝑟𝜃.
So now what we want to do is substitute in the values we have. So we have a value for the perimeter of the sector. And we have a value for the radius. So therefore, when we substitute in these values, we get 102 because that’s our perimeter minus — and then two multiplied by 27. And that’s because 27 is the radius. It’s gonna be equal to 27𝜃. And that’s because again the radius was 27.
So then, we’re gonna get 48. And that’s because it’s 102 minus 54, because two multiplied by 27 is 54. So 48 is equal to 27𝜃. So then, if we divide each side of the equation by 27, we can say that 48 over 27 is equal to 𝜃. Or if you flip it around the other way, 𝜃 is equal to 48 over 27.
But why is this going to be useful? Well, it’s going to be useful because we have another formula to help us work out the area of a sector. But to be able to use this formula for the area, we need to remember that the units of our 𝜃 or our angle are radians. And we can see that, here, I’ve shown that we’ve a small 𝑐 or an rad for radians. These are both notation that can be used to denote that it’s radians. And the formula for the area of a sector is that the area is equal to a half 𝑟 squared 𝜃. Again like we just said, 𝜃 must be in radians.
So therefore, substituting in the values that we have, we’ll have the area is equal to a half multiplied by 27 squared, and that’s because our radius is 27, then multiplied by 48 over 27 because that was the value of our 𝜃, which is gonna be equal to a half multiplied by 729. That’s because that’s 27 squared multiplied by 48 over 27, which will give a final area of 648 centimetres squared. And it’s centimetres squared because our radius was in centimetres.