Video: Pack 3 • Paper 2 • Question 13

Pack 3 • Paper 2 • Question 13

03:11

Video Transcript

𝐴𝐵𝑂 is a sector of a circle with center 𝑂 and radius 14 centimeters. Calculate the shaded area. Give your answer to one decimal place.

Now, in order to actually work out the area of the sector, so our shaded area, what we need to do first is actually find out what the angle, I put here as 𝑥, is because we’ll need that when we’re trying to calculate the area using the formula. We’ve actually got this relation that we know, which is that the angles around a point equals 360.

So we can use this to actually calculate what 𝑥 is gonna be because this tells us that if we have 𝑥 and we add it to 126, the answer is gonna be equal to 360 degrees. And therefore, if we actually subtract 126 from each side, what we’re gonna get is that 𝑥 is equal to 234 degrees. And now, I’ve added this to the diagram. So this is actually the angle of our sector.

Okay, so now, how we’re actually gonna work out what the area of the sector is going to be. Well, to work out the area of the sector, what we have is a formula to help us. And that formula is that the area of the sector is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared. So we actually get that formula because 𝜃 over 360 tells us the fraction of the circle. So it actually tells us what our sector is as a fraction of the whole entire circle. And 𝜋𝑟 squared is actually the area of the circle.

So what we’re saying is that the fraction of the circle multiplied by the area of the whole circle. And that will give us the area of this sector that we’re interested in. So therefore, what we’re gonna get is that the area of our sector is gonna be equal to 234 over 360. And that’s 234 because that’s the angle of our sector. So it’s a major sector in this case multiplied by 𝜋 multiplied by 14 squared.

And then, in next stage, if we’re gonna simplify, if we actually divide 234 and 360 by 18, we get 13 over 20. And then, this is multiplied by 14 squared which is 196. Then, multiply it by 𝜋. So we get 13 multiplied by 196 over 20𝜋, which when calculated will give us 400.2389 et cetera.

So okay, we’ve actually worked out the area of our shaded sector, but have we finished there? Well, if we check back to the question, what we want is actually our answer left to one decimal place. So therefore, we can say that the area of the shaded sector is gonna be equal to 400.2 centimeters squared to one decimal place.

And we get 400.2 because if we look back, if we want to one decimal place, I’ve put a line to the right of the one decimal place because that’s the two. And then, if we look at the number after it, that’s a three. And because that’s less than five, we’ll actually keep the two as it is without rounding up. So we’re left with 400.2 centimeters squared to one decimal place.

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