# 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,

**is the height, and**

*h***and**

*b*_{1}**are the lengths of the two bases.**

*b*_{2}

*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
** b_{1}** is equal to

**and that**

*10*meters**is**

*b*_{2}**.**

*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 metersNow, 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***, and**

*SA***are congruent to**

*AE***; they all have lengths of**

*RE***. Let's redraw our figure so that it displays the new information we've acquired.**

*5*centimeters

The right angles in the figure indicate that ** RS** and

**run perpendicular to each other. Therefore, we know that the perpendicular distance, or height, between**

*NM***and**

*RE***is**

*NM***.**

*5*centimeters
Now that we have the height of trapezoid ** REMN**, we just have to find
the length of this quadrilateral's second base,

**. In order to do this, we need to find the sum of segments**

*NM***,**

*NS***, and**

*SA***:**

*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

**. We get**

*REMN*

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

**. Let's see if this works.**

*42.5*
To find the area of ** ?RSN** we have

So, the area of ** ?RSN** is

**. Let's find the area of another figure inside of the trapezoid.**

*7.5*square centimeters
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

**We just have to find the area of the last triangle before we add the areas up.**

*25*square centimeters.
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