Determine, to the nearest tenth, the area
of the given figure.
We can see from the diagram that this
figure is a composite figure. That simply means that it’s made up of
two or more geometric figures. Let’s see if we can work out what these
different shapes are. Well, the three shapes at the top are
semicircles. Notice that these three semicircles are
all the same size or congruent. We can say this because the marking on
the radii shows that these are all the same lengths in the three semicircles. Looking at the shape on the lower part of
this figure, we could say that it’s composed of a rectangle and two triangles. However, there is a much easier
shape. And that is that we can identify that
this is a trapezoid.
And so to find the area of the entire
figure, we’ll need to find the areas of the three semicircles and the area of the trapezoid
and add them together. Let’s begin by finding the area of the
semicircles. We can recall that the area of a circle
is equal to 𝜋 times the radius squared. And so therefore, the area of a
semicircle would be half of that, 𝜋𝑟 squared over two. Before we use this formula, we need to
find out the value of the radius as this value of 10 inches refers to the diameter. As the radius is half of the diameter,
this means that our radius is five. And so we have the calculation 𝜋 times
five squared over two. We can write this as 25𝜋 over two. As this is an area, our units will be
squared, so we’ll have square inches.
We can keep our answer in terms of 𝜋 as
we’ll use it in the final calculation. But if we did change it into a decimal,
then we wouldn’t round that value yet. In a moment, we’ll be able to use this
value for the area of a semicircle to find the area of three semicircles. But let’s move on to finding the area of
the trapezoid. The area of a trapezoid is calculated by
𝑎 plus 𝑏 over two times ℎ, where 𝑎 and 𝑏 are the two parallel sides and ℎ is the
perpendicular height. Filling in the values then for our
trapezoid, we have a length of 20. And the other parallel length is formed
of three lots of 10 inches.
We can say this because we know that we
had three congruent semicircles that all had a diameter of 10 inches. And we also have a perpendicular height
of 20 inches. We then have a calculation of 50 over two
multiplied by 20, which simplifies to give us the answer of 500 square inches. We’ve now found enough information to
help us calculate the area of the entire figure. As we calculated that the area of one of
these semicircles is 25𝜋 over two, then as we established that our semicircles are
congruent. To find the area of three semicircles, we
would multiply the area of one semicircle by three. And to this we add the area of our
trapezoid, which was 500 square inches. We can use our calculator to evaluate
this as 617.8097245 and so on square inches.
But to complete our answer, we must round
to the nearest tenth. And as our second decimal digit is not
five or more, then our answer stays as 617.8 square inches. And so we found the area of this figure
by adding together the individual areas of each of the shapes within it.