In this video, we are going to look at how to apply the formulae for the area and the circumference of a circle to problems involving arc length and the area of a sector.
So, first, let’s make sure we understand what is meant by a sector and by this term arc length. So, a sector of a circle is part of a circle. And specifically, it’s part of a circle that is enclosed by two radii and then an arc that joins them. So, it’s a little bit like cutting a slice from a circular cake. When we talk about arc length, we mean the length of this curved part that I’ve labelled arc in each of these two diagrams. And when we talk about sector area, we mean the area within that portion of the circle.
So, let’s look at our first question. It asks us to calculate the area and the perimeter of the sector given in this diagram here.
Now, we can see that we’re given two pieces of information. We’re given the radius of the circle. And also importantly, we’re given the size of this angle here. We’re told that the angle made by those two radii is 120 degrees. Now, that’s really important because it tells us what portion of the circle we have in this sector here. So, what we’ll do is we’ll look at how to answer this specific question and then we’ll generalize from there so that we could work out the area and the perimeter of any sector where the angle at the centre is anything at all.
So, let’s look at the area first of all. So, if we recall our formula of the area of a circle is that the area is equal to 𝜋𝑟 squared. But we don’t have a full circle here. What we have is 120 degrees worth of the circle. Now, there’re 360 degrees in a full circle. And 120 degrees is a third of that. So, we must have a third of the area of the full circle. So, in order to work out the area of this sector, I’m gonna work out the area of a full circle with radius six centimetres and then I’m gonna divide it by three, as we only have a third of that circle.
So, here’s my calculation. The area of the sector is 𝜋 multiplied by six squared divided by three. And if I work that out as a multiple of 𝜋, then it’s 12𝜋 centimetres squared. And as a decimal, it’s 37.7 centimetres squared, rounded to one decimal place. So, there we have our method for calculating the area of this sector. We just did a third of the area of the full circle.
Now, let’s look at how to calculate the perimeter of this sector. So, the perimeter is the distance all the way around. And if you look at this sector, you’ll see that it’s composed of three parts. There’s the curved part, which is the arc. And then, there are two radii. So, in order to calculate the total perimeter, we need to add these three parts together. So, the perimeter is the arc length plus two 𝑟.
Now, in order to work out the arc length, we’ll use a similar idea. We want to work out the circumference of the full circle but then divide it by three, as we only have a third of that circumference. So, I’ve put the two formulae for the circumference of a circle, 𝜋𝑑, or two 𝜋𝑟, depending on whether you’re using the diameter or the radius. We have the radius, so I’ll use that second version.
So, this arc length is gonna be two multiplied by 𝜋 multiplied by six. But then, remember I’m gonna divide it by three, as I only have a third of the circumference. So, here’s my calculation, the arc length and then plus the two radii. So, that would give me an answer four 𝜋 plus 12 centimetres if I was giving my answer in terms of 𝜋. Or I can evaluate it as a decimal. And that gives me 24.6 centimetres to one decimal place. Remember to use the correct units for your answers here. So, when I was working out the area, my units were centimetres squared. And when I’m working out the perimeter, that’s a length, so my units are centimetres.
Now, let’s look at how we could generalize what we just did in the previous question so that we have formulae that will work for sectors with any radius and any angle at the centre. So, I’ve drawn a diagram of a sector here. I’ve labelled the radius as 𝑟 and the angle at the centre I’ve given the letter 𝜃 to, which we often use when we’re working with angles.
So, here are the formulae for circumference and area. Let’s think about how we would adapt them for just a sector of a circle. Let’s think about the arc length first of all. Now, in the previous example, we found the circumference of the full circle and then we divided it by three. The reason we divided it by three is because we had 120 degrees out of the full 360 degrees. So, we had a third of the circle. If we want to generalize that to any angle, then we just need to think about what fraction of the circle we have. So, we have 𝜃 degrees out of a possible 360, which means we have 𝜃 over 360, that fraction of the total circle.
Which means our formula of the arc length is this. 𝜋𝑑, which is the full circumference, but then we multiply it by 𝜃 over 360. So, we multiply it by the fraction of the circle we have, which could be a half or a third or a quarter or it could be something like 71 degrees out of 360. We could, of course, replace 𝑑 with two 𝑟 and have another version of this formula as well. Note, as well that in the previous example, we weren’t just asked to calculate the arc length, we were asked to calculate the full perimeter. In which case, we’d also need to add on twice the radius.
For the area of the sector, it’s exactly the same idea. We find the area of the full circle, 𝜋𝑟 squared. But then, we went to multiply it by the fraction of the circle that we actually have. So, that will give us this formula for the area of the sector, 𝜃 over 360 multiplied by 𝜋𝑟 squared. And we can use these two formulae, regardless of what the angle at the centre is. Even if it’s not a number that’s a nice factor of 360.
Now, there’s also a little bit of wording that you need to be familiar with, a particular way that questions about arc length or sector area could be phrased. And this phrase is, find the length of the arc subtended by an angle of 50 degrees, for example, at the centre of the circle. So, if you hear that phrase subtended, it just means the angle formed at the centre of the circle by these two radii. So, it’s what I’ve labelled as 𝜃 in the diagram here.
Okay, our final question in this video, given that the area of this sector is eight 𝜋 centimetres squared, calculate the total perimeter of the sector.
So, looking at the diagram, we can see that we’ve been given the radius of the circle, which is eight centimetres. But what we haven’t been given is the angle at the centre of the circle. We have, however, been given an extra piece of information. We know that the area of this sector is eight 𝜋 centimetres squared. So, you may want to pause the video and have a think about how you’re going to use this information.
What I’m going to do, first of all, is just add a little label to my diagram to call this unknown angle 𝜃. Now, if we knew that angle, we’d be able to work out the total perimeter using the methods we’ve described. Cause we’d be able to find the arc length, and we know what the radius is. So, what we need to do is work out what that angle at the centre of the circle is.
So, let’s think about how we can find that angle. We know that the formula for the area of sector is 𝜃 over 360 multiplied by 𝜋𝑟 squared. And we also know what the area of this sector is. So, we’ll be able to set ourselves up an equation, where we can substitute the value of eight for the radius, we can substitute the value of eight 𝜋 for the area, and then we can solve that equation in order to find what this angle at the centre is. So, here is that equation 𝜃 over 360 multiplied by 𝜋 multiplied by eight squared, that’s the radius squared, and it’s equal to eight 𝜋. So, this is the equation I need to solve in order to find the value of 𝜃.
So, steps to solve this equation, well, actually there’s 𝜋 on both sides. So, I can actually cancel a factor of 𝜋 from both sides. I’m effectively dividing both sides of the equation through by 𝜋. And this eight squared over 360, that’s 64 over 360, which as a simplified fraction becomes eight over 45. So, I have eight over 45 𝜃 is equal to eight. Now, I could continue from here in two ways probably. I could multiply by 45 and then divide by eight. Or seeing that I have a factor of eight on both sides, I could just cancel those through and then multiply by 45. And both of those methods will lead to the same result that 𝜃, this unknown angle, is equal to 45 degrees.
Now, that I know the angle, I can just work out the perimeter of the sector using the methods that we described in the previous example. So, the total perimeter will be the arc length plus twice the radius. So, the arc length using that formula 𝜃 over 360 times two 𝜋𝑟 will be 45 over 360 multiplied by two times 𝜋 times eight. And then, I’m adding twice the radius on. If I wanted to give my answer in terms of 𝜋 then, that simplifies to two 𝜋 plus 16 centimetres. Or I can work it out as a decimal. And it gives me this answer here of 22.3 centimetres. And that’s rounded to one decimal place.
So, just to recap what we did in this question. We knew the area. We had to work backwards from that in order to work out the angle at the centre. And then, from there, we were able to move on and calculate the total perimeter.
So, to summarise, we have looked at general formulae that we can use in order to work out the area or the perimeter of a sector or the arc length no matter what the radius and no matter what portion of the circle we are given.