- What is a Trapezoid?
- How to Find the Area of a Trapezoid
- Area of a Trapezoid Formula
- Area of a Trapezoid Examples

A **trapezoid** is a quadrilateral with *one* pair of parallel sides. So, this four-sided polygon is a plane figure and a closed shape. It has four line segments and four interior angles. The parallel sides are the trapezoid's two **bases**; the other two sides are its legs.

Usually the trapezoid is presented with the longer parallel side -- the **base** -- horizontal. A perpendicular line from the base to the other parallel side will give you the trapezoid's **height** or **altitude**.

In mathematics, an **average** is the sum of a group of numbers divided by the quantity of items in the group.

So if you have three people holding books, you can find the average number of books they are holding like this: Martin is holding 5 books, Mack is holding 3 books, and Maria is holding 4 books. Together, 12 books are being held by 3 people. So, 12 books ÷ 3 people = an average of 4 books each.

To find the area of a trapezoid, you will find the average length of the two bases.

To find the area of any trapezoid, start by labeling its bases and altitude. In our trapezoid, label the longer base $a$ and the shorter base $b$. Label a line perpendicular to the two bases $h$ for height or altitude of the trapezoid.

Notice we did not label the legs. We do not need to know anything about the length of the legs or the angles of the vertices to find area.

The formula for the area of a trapezoid is the average of the bases multiplied by the altitude. In the formula, the long and short bases are $a$ and $b$, and the altitude is $h$:

$area=\frac{a+b}{2}h$

Multiplying times $\frac{1}{2}$ is the same as dividing by 2. We take half the sum of the length of the two bases (their average) and then multiply that by the altitude, or height, to find the area in square units.

Trapezoid $LMNO$ has parallel bases $LM$ and $NO$. Line segment $LM$ is 7 cm long, and line segment $NO$ is 13 cm long. We will label longer side $NO$ as $a$ and shorter side $LM$ as $b$. The height, $h$, is 5 cm.

First, let's plug these numbers into our formula:

$area=\frac{13cm+7cm}{2}\times 5cm$

Next we add 13 plus 7 and get:

$area=\frac{20cm}{2}\times 5cm$

Then we divide by two, then and get:

$area=10cm\times 5cm$

Finally, we multiply and get our answer:

$area=50c{m}^{2}$

The area of this trapezoid is 50 square centimeters.

Now you try it! Another trapezoid has a long base $a$, 11 meters, and a shorter base $b$, 7 meters. Its altitude $h$ is 9 meters. What is the area in square meters?

$area=\frac{11cm+7cm}{2}\times 9cm$

Did you get 81 square meters? Your answer for area is always in square units of the linear measurement. So a trapezoid measured in feet gives an area in square feet, centimeters yield square centimeters, and so on.

Remember that multiplying by ½ is the same as dividing by 2, so you can add the lengths of the bases and then divide their sum by two, if that is easier for you.

Because of the commutative property of multiplication, you can rearrange these three numbers, $\frac{1}{2}$, altitude $h$, and the length of bases $a+b$, in any order to make the calculation easy.

So, with trapezoid $LMNO$, you could also have written the formula like:

$area=\frac{1}{2}\times 9\times (11+7)$

Here is one more example for you. The new trapezoid is upside down from how you usually see them, but don't let that stop you! The short base $b$ is 21 inches long. The long base $a$ (this time at the top of the drawing) is 31 inches long. The altitude $h$ (no matter which way you look at the trapezoid) is 5 inches.

$area=\frac{1}{2}\times 5\times (31+21)$

*OR*

$area=\frac{1}{2}\times (31+21)\times 5$

*OR*

$area=\frac{31+21}{2}\times 5$

However you use the formula, you will always get the same answer: $area=130i{n}^{2}$

In this lesson and video we have reviewed what a trapezoid is, examined how averages have a role in geometry, learned how to label and use the parts of a trapezoid to calculate area, and learned the formula for calculating the area of a trapezoid in square units.

After completing this lesson and studying the video, you should learn to:

- Recognize a trapezoid
- Apply the concept of averages to geometry
- Calculate the area of a trapezoid
- Recognize and apply the formula for finding area of a trapezoid

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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