Question Video: Finding the Area of Composite Figures Mathematics • 6th Grade

Determine the area of the shown figure, to the nearest tenth.


Video Transcript

Determine the area of the shown figure to the nearest tenth.

Let’s look carefully at the diagram. We can see, first of all, that we have a trapezoid or a trapezium. And from this trapezoid, a triangle has been removed to give the shaded region. The shaded area can, therefore, be calculated as the area of the trapezoid minus the area of the triangle.

We know the formulae for calculating the areas of each of these polygons, so we just need to determine their measurements from the diagram. Let’s consider the trapezoid first of all. 𝑎 and 𝑏 represent the two parallel sides of the trapezoid. We can see that one of the parallel sides is nine meters. And the other is the sum of the measurements of six meters, five meters, and six meters, which is 17 meters. We haven’t been given the perpendicular height of the trapezoid though. So, we’re going to need to find a way to calculate it.

In the triangle, we can see that the base is five meters. But, once again, we haven’t been given the perpendicular height. And I’m going to change the letters here so that we use capital 𝐻 to represent the height of the trapezoid and lowercase ℎ to represent the height of the triangle.

Let’s consider how we could calculate the height of this triangle. We can see from the diagram that it is an isosceles triangle because it has two equal sides each of nine meters. And so, we know that this perpendicular height divides the triangle into two identical right triangles. In each of these triangles, we know two side lengths. The length of nine meters, which is the hypotenuse as it’s opposite the right angle. And the length of 2.5 meters. That’s half the total base, five meters, of the original triangle.

As we know two of the side lengths and we wish to calculate the third, we can apply the Pythagorean theorem. Which tells us that in a right triangle, the square on the hypotenuse is equal to the sum of the squares of the two shorter sides, often written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared. We can, therefore, form an equation. ℎ squared plus 2.5 squared is equal to nine squared.

We can then subtract 2.5 squared from each side. And evaluating nine squared minus 2.5 squared gives ℎ squared equals 74.75. ℎ is, therefore, the square root of 74.75. And as a decimal, this is a little over 8.6. But we’ll keep this value for ℎ in its exact form for now.

So, now, we know the height of the triangle, we can work out its area. It’s a half multiplied by five multiplied by the square root of 74.75. Remember, we’re back in the full isosceles triangle here not the smaller right triangle. For the area of the trapezoid, we have a half the sum of the parallel sides. That’s a half multiplied by nine plus 17. And the total height, capital 𝐻, will be the value we found for the height of the triangle plus the additional two meters.

So, we now have a calculation we can evaluate to find the shaded area. We can work each area out separately, if we wish, and then subtract, giving 116.780. The question asked for the area to the nearest tenth. So, rounding our answer and including the units, we have that the area of the shown figure to the nearest tenth is 116.8 square meters.

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