9.4: Area of Parallelograms
Introduction
Go Wildcats!
One afternoon, Miguel’s assignment is to hang a banner in the stadium. The banner says “Go Wildcats” and is the shape of a parallelogram with a yellow background and dark blue border. He has two different places that he can put it. One spot is only 11 feet wide and one spot is 13 feet wide.
Miguel isn’t sure which spot is the best for the “Go Wildcats” banner. Wherever he chooses to hang the banner it needs to fit with a little space on each side so that people don’t bump it as they go past.
The banner is rolled up in a tube. Miguel looks at the label to see that the square footage of the banner is 42 square feet. The height of the banner is 3.5 feet. The length of the banner is missing. It seems to have been covered up with a shipping label.
Miguel is a bit frustrated. He isn’t sure that he has enough information to figure out the length of the banner. He knows that he needs to figure out the length of the banner in order to choose the correct spot to hang it. Miguel takes out a piece of paper to try to figure things out.
Does Miguel have enough information? Can he figure out the length of the banner using the given information? How long is the banner? Which spot is the best?
All of these questions can be answered by using parallelograms and this lesson will teach you all that you need to know to figure this one out.
What You Will Learn
In this lesson you will learn how to do the following:
- Recognize the formula for finding the area of a parallelogram.
- Find areas of parallelograms given base and height.
- Find unknown dimensions of parallelograms given area and one other dimension.
- Estimate actual areas of parallelograms in scale drawings.
- Solve real-world problems involving area of parallelograms.
Teaching Time
I. Recognize the Formula for Finding the Area of a Parallelogram
Remember back to our work on quadrilaterals? Well, here is one of the quadrilaterals that we learned about earlier, the parallelogram. A parallelogram is a four sided figure with opposite sides parallel. It doesn’t matter what the angles are in a parallelogram as long as the opposite sides are parallel.
If we wanted to figure out the distance around the edge of a parallelogram, then we would find the perimeter of the figure.
If we wanted to find the measure of the space inside the parallelogram, we would be finding the area of the parallelogram. This lesson is all about finding the area of parallelograms.
Let’s start off by looking at how we find the area of a common parallelogram, the rectangle. A rectangle is a parallelogram with four right angles. The opposite sides are parallel too.
To find the area of a rectangle, we multiply the length by the width.
\begin{align*}A=lw\end{align*}
Look at this example.
Example
To find the area of this rectangle, we use the formula and the given measurements.
\begin{align*}A&=lw\\ A&=(6 \ in)(2 \ in)\\ A&=12 \ square \ inches \ or \ in^2\end{align*}
Notice that the measurement is in square inches because inches \begin{align*}x\end{align*} inches is inches squared.
Write down the formula for finding the area of a rectangle in your notebook. Also be sure to include the statement in the box above.
Not all parallelograms have right angles. That is why some are called squares or rectangles and some are called parallelograms. The only necessary quality of a parallelogram is that the opposite sides need to be parallel.
How can we find the area of a parallelogram?
Because a parallelogram does not have right angles, multiplying the length and the width is not possible. Notice that the side of a parallelogram is at an angle. Because of this, we need to use a different measure to find the area of a parallelogram. We need to use the base and the height.
Notice that the base is the bottom measurement and the height is the measurement inside the figure.
When we multiply these two measurements, we can find the area of the parallelogram. Here is the formula.
\begin{align*}A=bh\end{align*}
Make a note of this formula in your notebooks.
II. Find Areas of Parallelograms Given Base and Height
Now that we know the formula for finding the area of a parallelogram, we can use it in problem solving. Look at this example.
Example
Find the area of the parallelogram below.
We can see that the base is 7 inches and the height is 3 inches. We simply put these numbers into the appropriate places in the formula and solve for \begin{align*}A\end{align*}.
\begin{align*}A &= bh\\ A &= 7 (3)\\ A &= 21 \ in.^2\end{align*}
Sometimes, you won’t have a picture to work with. When this happens, you can still find the area as long as you know the base and the height.
Example
Base = 5 inches, Height = 3.5 inches
\begin{align*}A&=bh\\ A&=(5)(3.5)\\ A&=17.5 \ in^2\end{align*}
9G. Lesson Exercises
Find the area for each parallelogram.
- Base = 9 ft., Height = 4 ft.
- Base = 7 meters, Height = 3.5 meters
- Base = 10 yards, Height = 7 yards
Check your answers with a peer. Did you label the measurement part of your answer correctly using square units?
III. Find an Unknown Dimension of a Parallelogram Given the Area and One Other Dimension
Sometimes, you will be given the area of the parallelogram and one other dimension such as the base or the height. Then you will have to use the formula and your problem solving skills to figure out the missing dimensions.
Example
A parallelogram has an area of \begin{align*}105 \ m^2\end{align*}. The height of the parallelogram is 7 m. What is its base?
In this problem, we know the area and the height of the parallelogram. We put these numbers into the formula and solve for the base, \begin{align*}b\end{align*}.
\begin{align*}A &= bh\\ 105 &= b(7)\\ 105 \div 7 & = b\\ 15 \ m & = b \end{align*}
By solving for \begin{align*}b\end{align*}, we have found that the base of the parallelogram is 15 meters. Let’s check our calculation to be sure. We can check by putting the base and height into the formula and solving for area
\begin{align*}A &= bh\\ A &= 15 (7)\\ A &= 105 \ m^2 \end{align*}
We know the area is \begin{align*}105 \ m^2\end{align*}, so our calculation is correct.
Let’s look at another one.
Example
The area of a parallelogram is 184 square yards and its base is 23 yards. Find its height.
This time we know the area of the parallelogram and its base. We can put these into the formula and solve for the height, \begin{align*}h\end{align*}.
\begin{align*}A &= bh\\ 184 &= 23h\\ 184 \div 23 &= h\\ 8 \ yd &= h \end{align*}
We have found that the parallelogram has a height of 8 yards. Again, let’s use the formula to check our work.
\begin{align*}A &= bh\\ A &= 23 (8)\\ A &= 184 \ yd^2\end{align*}
Our calculation is correct! Whenever we are given two pieces of information about a parallelogram, we can use the formula for area to find the third measurement.
9H. Lesson Exercises
Find the missing dimension in each parallelogram.
- Base = 8 inches, Area = 32 sq. inches. What is the height?
- Base = 9.5 inches, Area = 57 sq. inches. What is the height?
- Height is 2.5 ft, Area = 20 sq. feet. What is the base?
Take a few minutes to check your answers.
IV. Estimate Actual Areas of Parallelograms in Scale Drawings
Sometimes, you will see a scale drawing. A scale drawing is a drawing that uses a small measurement to represent a real-world measurement. We can work with a scale drawing and estimate actual areas of a parallelogram given the scale. Let’s look at an example.
Example
Estimate the area of this garden.
To work on solving this problem, we need to look at what information is given to us in the drawing.
We know that the scale says that 1” is equal to 3 feet.
The base of the garden in the drawing is 8” and the height is 3”.
We want to estimate the area.
Let’s start by estimating the base. \begin{align*}8 \times 3\end{align*}. We know that \begin{align*}8 \times 3\end{align*} is 24. This gives us a total of 24 feet for the base of the garden.
Let’s estimate the height. \begin{align*}3 \times 3\end{align*}. We know that \begin{align*}3 \times 3 = 9\end{align*}. The height of the garden is about 9 feet.
Next, we estimate the area of the garden.
24 rounds down to 20 and 9 rounds up to 10
\begin{align*}20 \times 10 = 200 \ square \ feet\end{align*}
How close it our estimate?
Let’s do the actual multiplying to figure this out.
\begin{align*}24 \times 9 = 216 \ square \ feet\end{align*}
We have an estimate that is reasonable.
V. Solve Real-World Problems Involving the Area of a Parallelogram
We have seen that we can apply the formula for finding the area of parallelograms 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 also use this formula when we are given real measurements. Let’s try a few problems involving parallelograms in the real world.
Example
Mrs. Vasquez is buying new carpet to redecorate her parallelogram-shaped store. To determine how much carpet she needs, she measured the length of the room and found it was 36 meters. Then she measured across the room with a line perpendicular to the first and found it was 16 meters across. How many square meters of carpet does Mrs. Vasquez need to buy?
Let’s begin by figuring out what the problem is asking us to find. We need to find how much carpet Mrs. Vasquez needs to cover the floor of her store, so we have to find the area of the store. That means we will use the area formula to solve for A. In order to use the formula, we need to know the base and height of the store.
We know that one side of the store will be 36 meters. Let’s call this the base. We also know that Mrs. Vasquez made a perpendicular line in order to measure the height, or the distance across the room. The height given in the problem is 16 meters. Let’s put this information into the formula and solve for area.
\begin{align*}A & = bh\\ A & = 36(16) \\ A & = 576 \ m^2\end{align*}
Mrs. Vasquez will need to buy 576 square meters of carpet.
Example
Christie is making a banner for the school talent show in the shape of a parallelogram. She used 59.5 square yards of paper. If the base of the banner is 17 yards, what is its height?
First, let’s make sure we understand the question in the problem. We need to find the height of the parallelogram. This time we are given the area and the base. As we’ve seen, we simply put these numbers into the formula and solve for the height, \begin{align*}h\end{align*}.
\begin{align*}A &= bh\\ 59.5 &= 17h\\ 59.5 \div 17 &= h\\ 3.5 \ yd &= h\end{align*}
The height of the banner is 3.5 yards.
As long as we have any two pieces of the area formula, we can solve for the third.
Now let’s look at applying what we have learned about parallelograms and area to solve the problem from the introduction.
Real–Life Example Completed
Go Wildcats!
Here is the original problem once again. Reread it and underline any important information.
One afternoon, Miguel’s assignment is to hang a banner in the stadium. The banner says “Go Wildcats” and is the shape of a parallelogram with a yellow background and dark blue border. He has two different places that he can put it. One spot is only 11 feet wide and one spot is 13 feet wide.
Miguel isn’t sure which spot is the best for the “Go Wildcats” banner. Wherever he chooses to hang the banner it needs to fit with a little space on each side so that people don’t bump it as they go past.
The banner is rolled up in a tube. Miguel looks at the label to see that the square footage of the banner is 42 square feet. The height of the banner is 3.5 feet. The length of the banner is missing. It seems to have been covered up with a shipping label.
Miguel is a bit frustrated. He isn’t sure that he has enough information to figure out the length of the banner. He knows that he needs to figure out the length of the banner in order to choose the correct spot to hang it. Miguel takes out a piece of paper to try to figure things out.
To start working on this problem, we can use the formula for finding the area of a parallelogram.
\begin{align*}A = bh\end{align*}
Next, we can fill in the area and the height. The base is the length of the parallelogram and that is unknown.
\begin{align*}42=b(3.5)\end{align*}
Now to solve for \begin{align*}b\end{align*}, we divide both sides by 3.5.
\begin{align*}42 \div 3.5 &= 12\\ b&=12 \ feet\end{align*}
The length of the banner is 12 feet. Given this length, Miguel should put it in the spot that is 13 feet wide.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Parallelogram
- a quadrilateral with opposite sides parallel.
- Perimeter
- the distance around a figure.
- Area
- the amount of space contained inside a two-dimensional figure.
- Scale Drawing
- a drawing of a life size image where the drawing is made smaller than the actual image using a scale.
Technology Integration
James Sousa, Khan Academy, Area of a Parallelogram
Other Videos:
- http://www.mathplayground.com/howto_sameareadiffperimeter.html – This is a basic video to help students understand the difference between area and perimeter.
Time to Practice
Directions: Find the area of each parallelogram given the base and the height.
1. Base = 9 inches, height = 5 inches
2. Base = 4 inches, height = 3 inches
3. Base = 12 feet, height = 6 feet
4. Base = 11 meters, height = 8 meters
5. Base = 13 yards, height = 10 yards
6. Base = 4 feet, height = 2.5 feet
7. Base = 5.5 inches, height = 3.5 inches
8. Base = 9 feet, height = 6.5 feet
9. Base = 22 miles, height = 18 miles
10. Base = 19 meters, height = 15 meters
Directions: Given the area and one dimension, find the missing dimension.
11. Base = 9 feet, area = 45 sq. ft.
12. Base = 8 in, area = 20 sq. in.
13. Base = 11 feet, area = 99 sq. feet
14. Base = 12 inches, area = 120 sq. inches
15. Base = 4.5 feet, area = 11.25 sq. ft.
16. Height = 3 inches, area = 36 sq. inches
17. Height = 4 feet, area = 72 sq. feet
18. Height = 5 meters, area = 80 sq. meters
Directions: Solve each problem.
19. A parallelogram has an area of 390 square centimeters. If its height is 15 cm, what is its base?
20. What is the height of a parallelogram whose base is 28 inches and area is 1,176 square inches?
21. Donna wants to cover her parallelogram-shaped crafts box in fabric. The base of the lid is 32.7 cm and the height is 12.2 cm. What is the area of the lid?
22. John is planting grass in a patch of lawn that is shaped like a parallelogram. The height of the parallelogram is 34 feet. The other border is 65 feet. How many square feet of grass will John plant?
23. Kara and Sharice are in a quilting competition. Both are stitching parallelogram-shaped quilts. So far Kara’s has an area of 2,278 square inches and a height of 44 inches. Sharice’s quilt has an area of 2,276 square inches and a height of 47 inches. Whose quilt is longer? By how many inches is it longer?
24. Denise bought a picture frame in the shape of a parallelogram. The area of the picture frame is 36,795 square centimeters. If its height is 165 centimeters, what is its base?