# Video: Pack 3 • Paper 3 • Question 17

Pack 3 • Paper 3 • Question 17

04:08

### Video Transcript

𝑂𝐴𝐵 is a sector of a circle with radius 12 centimetres. 𝐴𝐵 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 we’re actually gonna solve this problem, I’ve kind of shown using these little diagrams because what we’re gonna do is we’re gonna find the area of the total sector minus the area of the triangle 𝐴𝑂𝐵. And this will equal our shaded region. So what we’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 we’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 𝜃 over 360 multiplied by 𝜋𝑟 squared. And that’s where 𝜃 is the angle that’s within the sector. So what we’re saying is it’s a fraction of our circle multiplied by 𝜋𝑟 squared. Okay, so now what we’re gonna do is actually substitute our values in. And when we do that, we get 𝐴 is equal to 45 over 360. That’s because our angle is 45 degrees multiplied by 𝜋 multiplied by 𝑟 squared, so 12 squared.

Okay, great, now we can actually calculate this. So when we actually do calculate this, what we’re gonna get is that the area is equal to 144 over eight 𝜋. And we get that because we get 144 because 12 squared is 144 and then 45 over 360 gives us an eighth. So we’re left with 144 over eight 𝜋. So now, we can actually simplify this. And when we do, we get an answer of 18𝜋 centimetres squared. So great, we can say the area of the whole sector is 18𝜋 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 𝐴𝑂𝐵, we actually have a formula that can help us. And that formula is area of triangle is equal to a half 𝑎𝑏 sin 𝐶. And what this actually tells us is that the area is gonna be equal to a half multiplied by 𝑎𝑏, where 𝑎𝑏 are- are the two sides either side of the angle 𝐶. So it’s a half 𝑎𝑏 multiplied by sin 𝐶.

And in our triangle, 𝑎 and 𝑏 are both gonna be 12 centimetres because they’re the radius and the radius is either side of 45 degrees, which is 𝐶 our angle in the middle. So therefore, we can now substitute in the values 𝑎 equals 12, 𝑏 equals 12, and 𝐶 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 it’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 we’ll calculate this and when we do, we get 36 root two centimetres squared and that’s because 72 divided by two is 36. So we get 36 root two centimetres squared and that’s the area of the triangle 𝐴𝑂𝐵. 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 what’s happening, we’ve got 18𝜋 because that was the area of the whole sector minus 36 root two because that’s the area of the triangle 𝐴𝑂𝐵 which is gonna be equal to 5.6369 et cetera.

So it’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.