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 points 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.
To answer this, we’ll need to work out two things. The shaded area will be found by finding the area of the whole trapezium and then
taking away the area of the portion of the circle; that’s called a sector. So we’ll need to find the area of the trapezium and the area of a sector. Let’s begin with the trapezium and recall the formula for its area.
It’s a half multiplied by 𝑎 plus 𝑏 multiplied by ℎ, where 𝑎 and 𝑏 are the lengths
of the parallel sides and ℎ is the height between them. We are told that 𝑂𝐶 is a straight line. The lines 𝐴𝑂 and 𝐵𝐶 highlighted form right angles with the straight line
𝑂𝐶. This means that the lines 𝐴𝑂 and 𝐵𝐶 must be parallel since 90 plus 90 gives us
180 degrees and we know that supplementary angles add to 180 degrees. So the lengths of the parallel sides are four centimeters and seven centimeters.
The distance between them ℎ is the length of this side 𝑂𝐶. We know the length of part of this side. The length between 𝐶 and 𝐷 is six centimeters, but we can also find the length of
𝑂𝐷. 𝑂𝐷 is a line joining the centre of the circle to a point on its circumference. That means it’s actually the radius of the circle.
The radius of the circle is four centimeters. So the total length of 𝑂𝐶 can be found by adding four and six, which is 10
centimeters. Let’s substitute this in to the area of a trapezium. It’s a half multiplied by four plus seven multiplied by 10. And we can pop these numbers into our calculator and it gives us the area of the
trapezium to be 55 centimeters squared.
Next, we need to find the area of the sector of the circle. Notice that it’s made up of a right angle. We know that 90 degrees is one-quarter of a full turn. So this sector represents a quarter of the whole circle. To find its area then, we can find the area of the whole circle and divide it by
The area of a circle is 𝜋𝑟 squared, where 𝑟 is the radius. So to find the area of the sector, we can use the formula 𝜋𝑟 squared divided by
four. We already said that the radius of the circle is four. So this formula becomes 𝜋 multiplied by four squared all divided by four.
And if you pop that in your calculator, you might notice it gives the answer as four
𝜋. It’s absolutely fine to leave it in that form for a moment to prevent us needing to
use a long nasty number. We said that the shaded area is found by subtracting the area of the sector from the
area of the trapezium. So that’s 55 minus four 𝜋.
55 minus four 𝜋 is 42.433 and so on. We won’t round this number just yet. We are being asked to find the percentage of the trapezium that’s shaded. To do this, we need to divide the amount shaded by the total area of the
trapezium. And since it’s a percentage, we need to multiply this by 100.
Now, a lot of calculators will have a previous answer button that will allow us to
type this in exactly as we had it. It’s 42.433 and so on divided by 55, which was the total area of the trapezium,
multiplied by 100. That’s 77.152 and so on.
We need to round our answer correct to one decimal place. The first number after the decimal point is a one. So the number immediately to the right of that is its deciding digit. Since the deciding digit is five, that tells us to round our number up.
And we can see that the percentage of the trapezium that is shaded correct to one
decimal place is 77.2 percent.