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# 6.5: Area of Trapezoids

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Use formulas to find the area of trapezoids – quadrilaterals with exactly one pair of parallel sides.

## Area of a Trapezoid

Recall that a trapezoid is a quadrilateral with one pair of parallel sides. The lengths of the parallel sides are the bases. The perpendicular distance between the parallel sides is the height, or altitude, of the trapezoid.

In the trapezoid pictured above, the bases b1\begin{align*}b_1\end{align*} and b2\begin{align*}b_2\end{align*} are ______________________ to each other.

The altitude (or height) is _________________________ to both bases.

To find the area of the trapezoid, turn the problem into one about a parallelogram. Why? Because you already know how to compute the area of a parallelogram!

• Make a copy of the trapezoid.
• Rotate the copy 180\begin{align*}180^\circ\end{align*}.
• Put the two trapezoids together to form a parallelogram.

Two things to notice:

1. The parallelogram has a base that is equal to b1+b2\begin{align*}b_1 + b_2\end{align*}.

Look at the length of the base in the picture: the left side has a length of _______ and the right side has a length of _______ so the total length of the base of the parallelogram is b1+b2\begin{align*}b_1 + b_2\end{align*}.

2. The altitude (h)\begin{align*}(h)\end{align*} of the parallelogram is the same as the altitude (h)\begin{align*}(h)\end{align*} of the trapezoid.

Now to find the area of the trapezoid:

The area of the parallelogram is: A=basealtitude\begin{align*}A = \text{base} \cdot \text{altitude}\end{align*}.

Since the length of the base =\begin{align*}=\end{align*} ____________ and the altitude =\begin{align*}=\end{align*} _____ ,

Area of the parallelogram=(b1+b2)h\begin{align*}\text{Area of the parallelogram} = (b_1 + b_2) \cdot h\end{align*}

The parallelogram is made up of two congruent trapezoids, so the area of each trapezoid is half the area of the parallelogram.

The area of the trapezoid is half of (b1+b2)h\begin{align*}(b_1 + b_2) \cdot h\end{align*}:

Area of Trapezoid with Bases b1\begin{align*}b_1\end{align*} and b2\begin{align*}b_2\end{align*} and Altitude h\begin{align*}h\end{align*}

The bases of the trapezoid in the figure above are _______ and ________.

The altitude (or height) of the trapezoid is ______.

For a trapezoid with bases b1\begin{align*}b_1\end{align*} and b2\begin{align*}b_2\end{align*} and altitude h\begin{align*}h\end{align*},

A=12(b1+b2)h or A=(b1+b2)h2\begin{align*}A = \frac{1}{2} (b_1 + b_2) h \ \text{or} \ A = \frac{(b_1 + b_2) h}{2}\end{align*}

Notice that the formula for the area of a trapezoid could also be written as the “Average of the bases times the height.”

Example 1

What is the area of the trapezoid below?

Since this trapezoid is sideways compared to the ones you have seen so far in this lesson, the bases (which are parallel to each other) are on the right and left side of the shape. The altitude is horizontal instead of vertical.

Using the graph paper that the trapezoid above is on, you can count the boxes to determine the length of each important part of the shape.

You can see that the bases of the trapezoid are _____ and _____. The altitude is _____.

To find the area, multiply half of the sum of both bases by the altitude (or height):

A=12(b1+b2)h=12(4+6)3=12(10)3=53=15\begin{align*}A = \frac{1}{2} (b_1 + b_2) h & = \frac{1}{2} (4 + 6) \cdot 3\\ & = \frac{1}{2} (10) \cdot 3 = 5 \cdot 3 = 15\end{align*}

The area of the trapezoid is 15 square units.

1. How many bases does a trapezoid have?

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2. True or false: The bases of a trapezoid are perpendicular to each other.

3. In your own words, describe the altitude of a trapezoid:

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\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

4. True or false: The altitude of a trapezoid is parallel to the bases.

5. True or false:

The average of the bases is the same as the sum of the bases divided by 2.

6. Draw a picture of a trapezoid in the space below:

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\begin{align*}{\;}\end{align*}

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