Video: Pack 3 โ€ข Paper 3 โ€ข Question 17

Pack 3 โ€ข Paper 3 โ€ข Question 17

04:08

Video Transcript

๐‘‚๐ด๐ต is a sector of a circle with radius 12 centimetres. ๐ด๐ต is a chord of the sector. Calculate the area of the shaded section, giving your answer accurate to three significant figures.

Now, the way that weโ€™re actually gonna solve this problem, Iโ€™ve kind of shown using these little diagrams because what weโ€™re gonna do is weโ€™re gonna find the area of the total sector minus the area of the triangle ๐ด๐‘‚๐ต. And this will equal our shaded region. So what weโ€™re gonna need to do first of all is find out the area of the total sector and also find out the area of the triangle.

So weโ€™re gonna start by finding the area of the whole sector. Well, we actually have the formula to help us and that says that the area is equal to ๐œƒ over 360 multiplied by ๐œ‹๐‘Ÿ squared. And thatโ€™s where ๐œƒ is the angle thatโ€™s within the sector. So what weโ€™re saying is itโ€™s a fraction of our circle multiplied by ๐œ‹๐‘Ÿ squared. Okay, so now what weโ€™re gonna do is actually substitute our values in. And when we do that, we get ๐ด is equal to 45 over 360. Thatโ€™s because our angle is 45 degrees multiplied by ๐œ‹ multiplied by ๐‘Ÿ squared, so 12 squared.

Okay, great, now we can actually calculate this. So when we actually do calculate this, what weโ€™re gonna get is that the area is equal to 144 over eight ๐œ‹. And we get that because we get 144 because 12 squared is 144 and then 45 over 360 gives us an eighth. So weโ€™re left with 144 over eight ๐œ‹. So now, we can actually simplify this. And when we do, we get an answer of 18๐œ‹ centimetres squared. So great, we can say the area of the whole sector is 18๐œ‹ centimetres squared.

So now, what we can do is actually move on and find the area of the triangle. Well, in order to find the area of triangle ๐ด๐‘‚๐ต, we actually have a formula that can help us. And that formula is area of triangle is equal to a half ๐‘Ž๐‘ sin ๐ถ. And what this actually tells us is that the area is gonna be equal to a half multiplied by ๐‘Ž๐‘, where ๐‘Ž๐‘ are- are the two sides either side of the angle ๐ถ. So itโ€™s a half ๐‘Ž๐‘ multiplied by sin ๐ถ.

And in our triangle, ๐‘Ž and ๐‘ are both gonna be 12 centimetres because theyโ€™re the radius and the radius is either side of 45 degrees, which is ๐ถ our angle in the middle. So therefore, we can now substitute in the values ๐‘Ž equals 12, ๐‘ equals 12, and ๐ถ equals 45 into our formula. And when we substitute the values in, we get the area is equal to a half multiplied by 12 multiplied by 12 multiplied by sin 45. So then, we have this equal to 72 multiplied by root two over two. We get 72 because a half multiplied by 12 multiplied by 12. So itโ€™s a half multiplied by 144. So a half of 144 is 72. And then, we get root two over two because this is the exact value of sin 45. And so these are the ones we should actually remember.

Okay, so weโ€™ll calculate this and when we do, we get 36 root two centimetres squared and thatโ€™s because 72 divided by two is 36. So we get 36 root two centimetres squared and thatโ€™s the area of the triangle ๐ด๐‘‚๐ต. So now, the next stage is to actually subtract the area of our triangle from the area of the whole sector to actually find our area of our shaded section.

So now, if we come back to the diagram that I drew at the beginning just to help us understand whatโ€™s happening, weโ€™ve got 18๐œ‹ because that was the area of the whole sector minus 36 root two because thatโ€™s the area of the triangle ๐ด๐‘‚๐ต which is gonna be equal to 5.6369 et cetera.

So itโ€™s actually at this point what we do is we check what the question wants our answer left in. And we can see that it says that it wants it accurate to three significant figures. So therefore, we can say that the area of the shaded section is equal to 5.64 centimetres squared to three significant figures.

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