The diagram below shows a trapezium 𝐴𝐵𝐶𝑂, which overlaps a circle. Point 𝑂 is the centre of the circle. Point 𝐷 lies on the straight line 𝑂𝐶. Both point 𝐴 and 𝐷 lie on the edge of the circle. Calculate the percentage of the trapezium that is shaded. Give your answer to one decimal place.
So in order to actually solve this problem and find out the percentage of the
trapezium that is shaded, first of all, what we need to do is actually find out what
area is actually shaded. And to do this, this is gonna be equal to the area of the total trapezium minus the
area of quarter of the circle.
Well, what we’re gonna start with is finding the area of the total trapezium. And to do that, we have a formula. And that formula is that a half 𝑎 plus 𝑏 multiplied by ℎ because this is the area
of trapezium, where 𝑎 and 𝑏 are actually our parallel sides and ℎ is actually the
distance between them or the height.
So then, what we’re gonna do for ours is actually substitute in our values. So it’s gonna be a half multiplied by four plus seven because these are our parallel
sides. And then, this is gonna be multiplied by the height. And what’s the height? Well, if we take a look back at the diagram, we can see that 𝑂𝐷 is also gonna be
four centimetres because it’s a radius of the circle. So therefore, our height is gonna be 𝑂𝐶, which is gonna be four plus six which is
gonna be equal to 10 centimetres.
So therefore, we can say that the area of the trapezium is gonna be equal to a half
multiplied by four plus seven multiplied by 10, which is gonna give us 55
centimetres squared. Okay, great, so we’ve worked out the area of the trapezium.
Well, now, what we want to do is actually find out the area of our quarter
circle. And to help us do this, we actually have another formula. In this formula, is that the area of a circle is equal to 𝜋𝑟 squared. So therefore, the area of our quarter circle is gonna be equal to 𝜋 multiplied by
four squared. And that’s because four is our radius and then divided by four because as we said we
want a quarter of the circle. So we’re dividing it by four.
Well, therefore, what we’re gonna get is actually four 𝜋 centimetres squared. And that’s because we had 𝜋 multiplied by four squared which is 𝜋 multiplied by 16
which is 16𝜋. 16𝜋 divided by four is just gonna give us four 𝜋. I’ve left it in 𝜋 at this point just because I want to maintain the accuracy. So we’re gonna leave it as that cause we know that’s an exact value.
Okay, so now, what we can do is actually move on and find out the area of the shaded
part of our trapezium. So therefore, the area of the shaded region is gonna be equal to 55 because that was
the area of our total trapezium minus four 𝜋 because that was the area of our
quarter circle. And again, this would be centimetres squared. I’ve just put it in there just so we know what we’re dealing with. We don’t actually need it because we’re looking at percentage in this question. But it’s worth remembering units and what we’re working in. Okay, so we’re gonna leave it in this form, which is 55 minus four 𝜋 just again
because we want to maintain the accuracy.
So now, we’re gonna move on to the final part of the question. Well, for the final part of the question, what we want to do is actually calculate
the percentage of the trapezium that is shaded. And to actually calculate the percentage that’s going to be shaded, what we’re gonna
do is actually have the area of the shaded region which is 55 minus four 𝜋. Then, we divide that by the total area of the trapezium which is 55 and multiply it
by 100 to give us our percentage. And when we do this, we get 77.15 et cetera.
So have we finished yet? No, there’s one final stage because if we look at the question, it says give your
answer to one decimal place. So therefore, if we look at the first decimal place, it’s a one. The number after is five. So because it’s five or above, we round the one to a two. So therefore, we can say that the percentage of the trapezium that is shaded is 77.2