Video Transcript
The area of a circular sector is 815.1 centimeters squared and the central angle is 62 degrees. Find the arc length for the sector giving the answer to the nearest centimeter.
So, the first thing I wanted to do is just remind us what the difference between a sector and a segment are because this can sometimes confuse students. So, if we see here, we’ve got a sector. And this comes from the center. And then, we’ve got two radii out to the edge that forms the sector. Whereas a segment looks a little bit like you might get a segment from an orange.
So, in this question, what we’re looking at is a sector. And we know that the area is 815.1 centimeters squared. And the angle at the center is 62 degrees. And what we’re trying to find is the arc length, which I’ve marked on here as 𝑥.
Okay, great, so, have we got any formulae that can help us? Well, first of all, we know that the area of a sector is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared, where 𝜃 is the angle at the center. And the reason we get this is because 𝜃 over 360 gives us the fraction of the circle that we’ve got. And then, 𝜋𝑟 squared is the area of a circle.
Okay, so let’s use this first to see if we can work out what we’ve got missing. But what do we have missing? Well, what we got missing is the radius. So, we need to try and find out the radius. But why do we need the radius? Well, we want the radius because, ultimately, what we’re trying to find is the arc length. And if we’ve got formula for arc length. It is for the arc length is equal to 𝜃 over 360 multiplied by two 𝜋𝑟. We could also have multiplied by 𝜋𝑑. It’s the same thing because we know that the radius is half the length of the diameter.
Okay, great, so, we can see what we’ve got, and we can see what we need to find. Let’s get on and find the radius. So, using our first formula, when we substitute in our values, we’re gonna get 815.1 equals 62 over 360 multiplied by 𝜋𝑟 squared. So now, what I’m gonna do is divide both sides of the equation by 62 over 360. I’m keeping it in the fraction form because we want to maintain accuracy all the way through till the end. So now, we’ve got 815.1 over 62 divided by 360 equals 𝜋𝑟 squared.
So, if we calculate what we have on the left-hand side using a calculator, what we’re gonna get is. Again, I’ve left it here in fractional form not decimal form. And that’s because I want to avoid rounding at this stage. So, what we get is 146718 over 31 is equal to 𝜋𝑟 squared. And then, dividing through by 𝜋, we get 1506.509351 is equal to 𝑟 squared. So then, if we square root both sides of the equation, we’re gonna get 38.81377785 is equal to 𝑟. As you see here, I’ve again written down everything that’s on the calculator display because we’re trying to maintain the accuracy so that we don’t get any accuracy errors before the end.
Okay, so, now we found our radius, what we want to do is now use this to find out our arc length. So now, to find the arc length, if we use our second formula, which is arc length is equal to 𝜃 over 360 multiplied by two 𝜋𝑟. Well, if we substitute in our values, we get 𝑥, because that was our arc length, is equal to 62 over 360. Multiplied by two multiplied by 𝜋 then multiplied by our 𝑟, which is 38.8 continued. Which is gonna give us a value of 𝑥 or an arc length of 42.00054951.
Well, if we take a look at the question, we could see that the question wants us to leave our answer to the nearest centimeter. So therefore, we can say that the arc length to the sector to the nearest centimeter is 42 centimeters.
