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
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.