Areas of Trapezoids

Recall that a
trapezoid is a
quadrilateral defined by one pair of opposite sides that run parallel
to each other. These sides are called bases, whereas the opposite sides that
intersect (if extended) are called legs. Let’s learn how to measure the areas
these figures.

Determining the area
of a trapezoid is reliant on two main components of these polygons: their
bases and heights. These characteristics helped us find the areas of parallelograms
and triangles in the previous section, but there is a slight difference
in finding the area of trapezoids: we require the measure of both of its bases.
This was not a requirement for parallelograms, and even if it were, we would know
their measures since a parallelogram’s bases are congruent.

Let’s begin studying the area of a trapezoid. The area of a trapezoid is equal to
one half the height multiplied by the sum of the lengths of the bases. It is expressed
as

where A is the area of the trapezoid, h is the height,
and b₁ and b₂ are the lengths
of the two bases.

The bases and height of the trapezoid are required in order to determine its area.

Let’s work on two exercises that will help us apply this area formula to trapezoids.

Exercise 1

Find the area of trapezoid ABCD.

Solution:

This problem appears to be quite simple because we are given the lengths of both
bases and the height of the trapezoid. It does not matter which base we choose as
our first or second base (because addition is commutative). We will just say that
b₁ is equal to 10 meters and that b₂
is 18 meters.

The height of our trapezoid is the perpendicular distance between our bases. The
illustration shows that this distance is equal to 9 meters. Now that
we have the measures of both bases and the height, we can plug them into the area
formula for trapezoids. We have

So, the area of trapezoid ABCD is 126 square meters.

Now, let’s try an exercise that requires a bit more work than the first problem.

Exercise 2

Find the area of trapezoid REMN.

Solution:

Finding the area of trapezoid REMN will require some initial work
because we are not given the length of both bases or the height of the figure. Let’s
use the properties we know about quadrilaterals to help us deduce some important
information.

Notice that there are tick marks around quadrilateral REAS. This means
that all sides of the quadrilateral are congruent. So, we know that segments RS,
SA, and AE are congruent to RE; they all
have lengths of 5 centimeters. Let’s redraw our figure so that it
displays the new information we’ve acquired.

The right angles in the figure indicate that RS and NM
run perpendicular to each other. Therefore, we know that the perpendicular distance,
or height, between RE and NM is 5 centimeters.

Now that we have the height of trapezoid REMN, we just have to find
the length of this quadrilateral’s second base, NM. In order to do
this, we need to find the sum of segments NS, SA, and
AM:

We see that our second base has a length of 12 centimeters. Now, we
are ready to plug our values into the area formula to find the area of trapezoid
REMN. We get

The area of trapezoid REMN is 42.5 square centimeters.

Alternate Solution:

Is there another way to solve this problem to assure ourselves that our solution
is correct?

The answer is yes. Notice that we can split up trapezoid REMN into
two triangles and a square. Therefore, if we take the sum of their areas, they should
add up to 42.5. Let’s see if this works.

To find the area of ?RSN we have

So, the area of ?RSN is 7.5 square centimeters. Let’s
find the area of another figure inside of the trapezoid.

We know that quadrilateral REAS is a parallelogram. In fact, it is
a square because it has four congruent sides and four right angles. We find this
area by doing the following:

We see that quadrilateral REAS has an area of 25 square centimeters.
We just have to find the area of the last triangle before we add the areas up.

The last triangle, ?EAM, is determined by performing the following
steps:

So, ?EAM has an area of 10 square centimeters.

Finally, we take the sum of these three polygons which make up the trapezoid. We
get

Indeed, we are correct about trapezoid REMN having an area of 42.5
square centimeters.