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9.6: Area of Trapezoids

Created by: CK-12

Introduction

The Bridge

Every day when Miguel rides his bike to the ball field at the University, he rides over a bridge. Since he sees it everyday, he usually doesn’t even think about it. He just rides his bike and ends up at the University.

One day when Miguel arrived at the bridge there were some workers repairing part of it.

“What are you doing?” he asked.

“We’re restoring this side of the bridge. You know, this is a famous kind of bridge design called a “Whipple Truss Bridge” it is in the shape of a trapezoid,” the first man said.

“Wow, I didn’t know that. How long is it?”

“It is 300 feet long, 40 feet high and the top length is 270 feet.”

“That is pretty cool. Well I have to go. See you later,” Miguel said, riding off.

He kept thinking about that bridge the rest of the day. Later that week, Miguel went to the library and took out a book on the Whipple Truss Bridge and other types of bridges. He has decided to do some investigating.

If the workmen were fixing one side of the bridge, what is the area that they are fixing? To figure this out, you will need to know how to find the area of a trapezoid. This lesson will teach you everything that you need to know about finding the area of a trapezoid.

What You Will Learn

By the end of this lesson, you will be able to complete the following:

  • Recognize the formula for area of a trapezoid.
  • Find the area of a trapezoid given the bases and the height.
  • Find unknown dimensions of a trapezoid given the area and one other dimension.
  • Find areas of combined figures involving trapezoids.

Teaching Time

I. Recognize the Formula for Area of a Trapezoid

A trapezoid is a quadrilateral with special properties. A trapezoid is a four sided figure with one pair of opposite sides parallel. These parallel sides are called the bases of the figure.

Look at this picture of a trapezoid.

You can see the bases in the picture. We can also see the height is marked. But what else do you notice about a trapezoid?

A trapezoid can be divided into two triangles.

Now you may wonder why that is at all important. It is important because if we want to find the area of a trapezoid, then we have to take each part of the trapezoid into consideration.

Remember that area is the amount of space inside a two-dimensional figure.

We can also see the two triangles here. Because the bases are different, there are two different triangles pictured here.

How does this all go together?

As with triangles, one of the dimensions we use to find the area of a trapezoid is its height. The other dimension we use is the base. However, a trapezoid has two bases. We can think of them as the bases of the two triangles within the trapezoid. Therefore we can use the area formula for triangles to help us find a formula for the area of trapezoids. Think about one triangle at a time. Here is the formula.

A  =  \frac{1}{2}bh

If we use this formula to represent each triangle within the trapezoid, we can write formula to represent the entire trapezoid.

A  =  \frac{1}{2} b_1 h   + \frac{1}{2} b_2 h

In the formula, b_1 stands for the base of one triangle and b_2 represents the base of the second triangle. We use h to represent the height of each triangle, or the height of the trapezoid.

We can combine these and write a simpler formula.

A  = \frac{1}{2} (b_1 +  b_2)h

We can also think of b_1 and b_2 as the parallel sides of the trapezoid. Remember, though, that h is NOT simply one of the slanted sides. The height of a trapezoid is always perpendicular to the bases. As long as we are given the measures of the bases and the height of a trapezoid, we can use this formula to solve for the area, no matter what shape or size the trapezoid is. Now let’s see how this works.

Take a few minutes to write this formula down in your notebook.

II. Find the Area of Trapezoids Given Bases and Height

Now that you understand the formula and where it comes from, we can look at applying it to finding the area of a trapezoid.

Example

Find the area of the trapezoid below.

We can see that the one base is 11 centimeters, one is 5 centimeters, and the height is 6 centimeters. We simply put these numbers into the appropriate places in the formula and solve for area, A.

A & =  \frac{1}{2}(b_1 + b_2)h\\A  & =  \frac{1}{2} (11  +  5) (6) \\A  & = \frac{1}{2} (16) (6)\\A  & = 8 (6)\\A  & = 48 \ cm^2

The area of this trapezoid is 48 square centimeters.

Example

What is the area of the trapezoid below?

Again, we simply put the information we have been given into the appropriate places in the formula and solve for A, area.

A  & =  \frac{1}{2}(b_1 + b_2)h\\A  & = \frac{1}{2} (9 + 14) (17)\\A  & = \frac{1}{2} (23) (17)\\A  & = 11.5 (17)\\A  & = 195.5 \ in.^2

The area of this trapezoid is 195.5 square inches.

9K. Lesson Exercises

Find the area of each trapezoid.

Take a few minutes to check your answers with a friend.

III. Find Unknown Dimensions of a Trapezoid Given Area and another Dimension

Sometimes a problem will give us the area and ask us to find one of the dimensions of the trapezoid—one of its bases or its height. We simply put the information we know in for the appropriate variable in the formula and solve for the unknown measurement. Let’s try an example.

Example

A trapezoid has an area of 22.5 \ m^2. The bases of the trapezoid are 6 m and 9 m. What is its height of the trapezoid?

In this problem, we know the area and the bases of the trapezoid. We put these numbers into the formula and solve for the height, h.

A  & =  \frac{1}{2}(b_1 + b_2)h\\22.5  & =  \frac{1}{2} (6  +  9) h\\22.5  & =  \frac{1}{2} (15) h\\22.5  & =  7.5h\\22.5  \div  7.5 & =  h\\3 \ m  & =  h

By solving for h, we have found that the height of the trapezoid is 3 meters.

Let’s check our calculation to be sure. We can check by putting the bases and height into the formula and solve for the area that we started with in the original problem.

A  & = \frac{1}{2} (b_1 + b_2)h\\A  & =  \frac{1}{2} (6  +  9) (3)\\A  & =  \frac{1}{2} (15) (3)\\A  & =  7.5 (3)\\A  & =  22.5 \ m^2

We know from the problem that the area is 22.5 \ m^2, so our calculation is correct.

Yes, you could say that. But sometimes we have to a few more steps than this to figure out the missing dimensions. Look at this example.

Example

The area of a trapezoid is 98 square yards. Its height is 7 yards and one of its bases is 16 yards. Find the measure of its other base.

This time we know the area of the trapezoid, its height, and one of its bases. We can put these into the formula and solve for the second base, b_2.

A  & = \frac{1}{2} (b_1 +  b_2)h\\98  & =  \frac{1}{2} (16  +  b_2) (7)\\98 \div 7  & = \frac{1}{2} (16  +  b_2) \qquad \text{Divide both sides by} \ 7.\\14  & = \frac{1}{2} (16  +  b_2)\\14  & = \frac{1}{2} (16)  +  \frac{1}{2} (b_2) \ \ \text{Use the Distributive Property to multiply by} \ \frac{1}{2}\\14  & =  8  +  \frac{1}{2} (b_2)\\14  -  8  & =  \frac{1}{2} (b_2) \qquad \qquad \ \text{Subtract} \ 8 \ \text{from both sides.}\\6  & = \frac{1}{2} (b_2)\\6 \times 2  & =  b_2\\12 \ yd  & =  b_2

Whew! Let’s walk through what we did!

Remember that we need to isolate b_2. To do so, we perform inverse operations on each side of the equation. First we divided both sides by 7. Then we subtracted 8 from both sides. By isolating b_2 we found that the second base of the trapezoid is 12 yards.

Again, let’s use the formula to check our work.

A  & =  \frac{1}{2} (b_1 +  b_2)h\\A  & =  \frac{1}{2} (12  +  16) (7)\\A  & =  \frac{1}{2} (28) (7)\\A  & =  14 (7)\\A  & =  98 \ yd^2

Our calculation is correct!

Whenever we are given information about a trapezoid, we can use the formula for area to find the unknown measurement.

IV. Find Areas of Combined Figures Involving Trapezoids

We mentioned that we can use trapezoids to find the area of larger figures. Remember that we could divide a trapezoid into two triangles? If we know the area of the triangles, we can add their areas together to find the area of the trapezoid.

We can do this for all kinds of figures. If we can divide the figure into trapezoids, for example, we can find the area of each trapezoid and add the areas together.

Also, we have seen that we can apply the formula for finding the area of trapezoids to different kinds of situations. Sometimes we need to solve for the area, but other times we may need to find the height or the base. We can use information given about a larger figure whenever that information corresponds to the height, bases, or area of one of the trapezoids contained within it. Let’s try an example to get a better idea of how this works.

Example

Find the area of the figure below.

We need to find the area of the whole figure. We can see that this hexagon is made up of two congruent trapezoids (we know they are congruent because the line divides the hexagon into two equal halves). If we can find the area of each of these, we can add them together to find the area of the whole figure. Let’s give it a try.

The height of each trapezoid is 5 inches and the longer base is 20 inches. What is the length of the shorter base? Look carefully. It is 13 inches. We can use these measurements to solve for the area of each trapezoid.

A  & =  \frac{1}{2} (b_1 + b_2)h\\A  & = \frac{1}{2} (13  +  20) (5)\\A  & = \frac{1}{2} (33) (5)\\A  & =  16.5 (5)\\A  & =  82.5 \ in.^2

Great! Now we know that each trapezoid has an area of 82.5 square inches. This is exactly half the area of the hexagon. We need to add the area of the two trapezoids together to find the area of the hexagon. Or, because the trapezoids have the same area, we can multiply by 2.

82.5 \ in.^2   +   82.5 \ in.^2     =   165 \ in.^2

This is our answer.

Anytime you are working with figures that are combined, we can break them down into simpler parts, find each area and then find the sum of the areas!!

Now let’s go back to the problem in the introduction!

Real–Life Example Completed

The Bridge

Here is the original problem once again. Reread it and underline any important information.

Every day when Miguel rides his bike to the ball field at the University, he rides over a bridge. Since he sees it everyday, he usually doesn’t even think about it. He just rides his bike and ends up at the University.

One day when Miguel arrived at the bridge there were some workers repairing part of it.

“What are you doing?” he asked.

“We’re restoring this side of the bridge. You know, this is a famous kind of bridge design called a “Whipple Truss Bridge” it is in the shape of a trapezoid,” the first man said.

“Wow, I didn’t know that. How long is it?”

“It is 300 feet long, 40 feet high and the top length is 270 feet.”

“That is pretty cool. Well I have to go. See you later,” Miguel said, riding off.

He kept thinking about that bridge the rest of the day. Later that week, Miguel went to the library and took out a book on the Whipple Truss Bridge and other types of bridges. He has decided to do some investigating.

To find the area of the trapezoid of the bridge, we need the dimensions for one side. The bases are 300 feet and 270 feet. The height is 40 feet.

Now we can substitute these values into our formula and solve.

A & = \frac{1}{2} (base+base)h\\A&=\frac{1}{2}(300+270)(40)\\A&=\frac{1}{2}(570)(40)\\A&=\frac{1}{2}(22,800) \\A&=11,400

The area of one side of the bridge is 11,400 square feet.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Trapezoid
a figure with four sides and one pair of opposite sides parallel.
Area
the space inside a two-dimensional figure.
Bases
of a trapezoid are the two parallel sides
Height
the measurement inside the trapezoid from base to base

Technology Integration

Khan Academy, Area of Trapezoids

Other Videos:

  1. http://www.mathplayground.com/howto_area_trapezoid.html – This is a great video that teaches you how to find the area of a trapezoid.

Time to Practice

Directions: Given the bases and the height, find the area of each trapezoid.

1. Base = 12 in, Base = 8 in, Height = 6 in

2. Base = 10 ft, Base = 6 ft, Height = 4 ft.

3. Base = 14 m, Base = 12 m, Height = 10 m

4. Base = 16 m, Base = 14 m, Height = 12 m

5. Base = 8 in, Base = 10 in, Height = 6 in.

6. Base = 7 ft., Base = 11 ft, Height = 5 ft.

7. Base = 9 ft., Base = 7 ft, Height = 6 ft.

8. Base = 4 in, Base = 6 in, Height = 3 in

9. Base = 6 in, Base = 5 in, Height = 2.5 in

10. Base = 7 in, Base = 9 in, Height = 3.5 in

Directions: Find the missing dimension in each example.

11. Base = 10 in, Base = 8 in, Area = 36 sq. in, What is the height?

12. Base = 12 m, Base = 10 m, Area = 88 sq. m. What is the height?

13. Base = 14 ft, Base = 12 ft, Area = 99 sq. ft. What is the height?

14. Base = 5 m, Base = 3 m, Area = 8 sq. m What is the height?

15. Base = 8 ft, Base = 6 ft, Area = 17.5 sq. ft. What is the height?

16. Base = 11 m, Base = 9 m, Area = 60 sq. m. What is the height?

Directions: Solve each problem.

17. Julius drew a trapezoid that had bases of 15 and 11 inches and a height of 4 inches. What is the area of the trapezoid Julius drew?

18. What is the height of a trapezoid whose bases are 14 inches and 19 inches and area is 99 square inches?

19. A trapezoid has an area of 247.5 square centimeters. If one base is 20 cm and its height is 9 cm, what is the measure of its other base?

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