10.1: Area of Parallelograms
Introduction
The Quilt Block
Jillian’s grandmother is coming to spend the summer with Jillian and her family. Jillian is very excited. Not only does Jillian love talking and visiting with her grandmother, but she loves to watch her sew. Jillian’s grandmother is a quilter and has been for some time. When Jillian visited during the holidays she told Jillian that she would help her make a quilt over the summer. Jillian can hardly wait to begin!!
The day after Jillian’s grandmother arrived, she and Jillian began planning for the first square of Jillian’s quilt. Her grandmother has selected a 12” quilt square for Jillian to start with. The square is made up of parallelograms and right triangles.
Jillian knows about parallelograms from school, but transferring the information to the quilt square has her puzzled. Here is a picture of the square that Jillian is going to make.
The quilt square is made up of 8 parallelograms. Each one has a base length of \begin{align*}3 \frac{1}{2}\end{align*} inches, sides 3 inches long, and a height of 2 inches.
Here is a picture of what one looks like.
Jillian needs to figure out the area of each parallelogram and then multiply that number by 8 so she will know how much material she will need for the 8 parallelograms in this first square.
Jillian is puzzled. She can’t remember how to figure this out. She knows that the area of a parallelogram is related to the area of a rectangle, but she can’t remember how to connect them. This is where you can help-this lesson will teach you how to help Jillian. Pay close attention and we will come back to this problem at the end of the lesson.
What You Will Learn
By the end of this lesson, you will learn the following skills:
- Recognize the area of a parallelogram as the area of a related rectangle.
- Find the area of parallelograms given base and height.
- Find unknown dimensions of parallelograms given area and another dimension.
- Estimate actual areas approximated by parallelograms in scale drawings.
- Solve real world problems involving area of parallelograms.
Teaching Time
I. Recognize the Area of a Parallelogram as the Area of a Related Rectangle
In the introduction problem, Jillian knew that the area of a parallelogram was related to the area of a rectangle but she couldn’t remember how to make the connection. Let’s begin by looking at the area of a rectangle and then see if we can connect this to the area of a parallelogram.
First, what is area?
Area is the space that is contained within the perimeter of a shape. When we talk about area we are referring to the surface or covering of something.
How do we find the area of a rectangle?
To find the area of a rectangle, we need to find the measurement for the inside of a rectangle.
Here is a rectangle. It has a width and a length. We can find the area of a rectangle by multiplying the length times the width.
5 \begin{align*}\times\end{align*} 10 \begin{align*}=\end{align*} 50 square inches
Notice that we multiplied units, so our answer is in square inches not just inches.
Many of us remember how to do this with a little review. Now let’s relate this to finding the area of a parallelogram.
We can look at the area of the rectangle in square units.
This rectangle is 18 units. We multiply the width of two times the length of 9 and get the area of 18 square units.
Next, we look at a parallelogram.
If we take off the two triangles on the ends, you can see that the parallelogram is a lot like the rectangle.
The big difference is in the angles. The rectangle has right angles in it so multiplying by the length and the width is challenging. The parallelogram has a height instead.
In this parallelogram we have a base of 8 and a height of 2.
8 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 16
The area of the parallelogram is 16 square units.
How did we do that?
Well, in figuring out the area of a rectangle, we multiplied the length times the width.
To find the area of a parallelogram, we multiply the base times the height.
II. Find the Area of a Parallelogram Given Base and Height
In the last section, you could see that while a rectangle and a parallelogram are related, that the parallelogram doesn’t really have a width that you can easily measure. Because a parallelogram has a side that is on an angle other than a 90 degree angle, we have to calculate the height of the parallelogram and use that as our width.
If we have the base of the parallelogram and its height, we can figure out the area of the parallelogram. We multiply one by the other. In this way, the formula is very much like the one we use for rectangles, where we multiply the length times the width.
Let’s look at an example.
Example
To find the area of this parallelogram, we multiply the base times the height.
\begin{align*}A & = bh\\ A & = (3)(8) \\ A & = 24 \ sq. \ inches\end{align*}
Practice a few of these on your own. Find the area of the following parallelograms.
- Base = 9 inches, Height = 3.5 inches
- Base = 5 inches, Height = 3 inches
Take a few minutes to check your work with a peer. Did you remember to label your work in square inches or centimeters? Don’t let the decimals trip you up!!
III. Find Unknown Dimensions of Parallelograms Given Area and Another Dimension
We can also work to figure out a missing dimension if we have been given the area and another measurement.
We can be given the area and the height or the area and the base.
This is a bit like being a detective. You will need to work backwards to figure out the missing dimension.
Let’s look at figuring out the base first.
Example
A parallelogram has an area of 48 square inches and a height of 6 inches. What is the measurement of the base?
To figure this out, let’s look at what we know to do. The area of a parallelogram is found by multiplying the base and the height. If we are looking for the base or the height, we can work backwards by dividing.
We divide the given area by the given height or given base.
48 \begin{align*}\div\end{align*} 6 \begin{align*}=\end{align*} 8
The measurement of the base is 8 inches.
This will work the same way if we are looking for the height. Let’s look at another example.
Example
A parallelogram has an area of 54 square feet and a base of 9 feet. What is the height of the parallelogram?
We start by working backwards. We get the area by multiplying, so we can take the area and divide by the given base measurement.
54 \begin{align*}\div\end{align*} 9 \begin{align*}=\end{align*} 6
The measurement of the height is 6 feet.
Practice a few of these on your own. Find the missing height or base using the given measurements.
1. Area = 25 square meters
Base = 5 meters
2. Area = 81 square feet
Base = 27 feet
3. Area = 36 square inches
Height = 2 inches
Take a few minutes to check your work. Did you find the correct height or base measurement?
IV. Estimate Actual Areas Approximated by Parallelograms in Scale Drawings
A scale drawing is a drawing that has a measurement with a relationship to the actual dimensions of something. For example, if we wanted to design a 50 foot tall building, we wouldn’t draw our design as actually 50 feet. Think about how huge the paper would be!!
Instead, we use a scale. Let’s say we use \begin{align*}\frac{1}{2}\end{align*}” for every foot, well now our drawing would be 25 inches tall and that is very useable for a design.
What does this have to do with parallelograms?
Well, sometimes, you can use a parallelogram to figure out an approximate distance or dimension. Let’s look at an example.
Example
Let’s say that we wanted to use this map and the parallelogram to estimate the area in and around Berlin.
Using the scale, we could say that the base of the parallelogram is \begin{align*}1\frac{1}{2}\end{align*}”. Therefore, we could estimate that the length of the base is 24 miles.
The height is probably about an inch. Therefore, the height is 16 miles.
Next, we multiply.
24 \begin{align*}\times\end{align*} 16 \begin{align*}=\end{align*} 384 square miles
We can see that our estimate for the area within the parallelogram is approximately 384 square miles.
Note: The actual area of Berlin is 344.4 sq. miles according to www.wikipedia.org.
This strategy of estimation can be used on maps, floor plans or buildings.
Real Life Example Completed
The Quilt Block
Remember Jillian and her quilt square? Here is the problem once again. Reread it and underline any important information.
Jillian’s grandmother is coming to spend the summer with Jillian and her family. Jillian is very excited. Not only does Jillian love talking and visiting with her grandmother, but she loves to watch her sew. Jillian’s grandmother is a quilter and has been for some time. When Jillian visited during the holidays she told Jillian that she would help her make a quilt over the summer. Jillian can hardly wait to begin!!
The day after Jillian’s grandmother arrived, she and Jillian began planning for the first square of Jillian’s quilt. Her grandmother has selected a 12” quilt square for Jillian to start with. The square is made up of parallelograms and right triangles.
Jillian knows about parallelograms from school, but transferring the information to the quilt square has her puzzled. Here is a picture of the square that Jillian is going to make.
The quilt square is made up of 8 parallelograms. Each one has a base length of \begin{align*}3\frac{1}{2}\end{align*} inches, sides 3 inches long, and a height of 2 inches.
Here is a picture of what one looks like.
Jillian needs to figure out the area of each parallelogram and then multiply that number by 8 so she will know how much material she will need for the 8 parallelograms in this first square.
Jillian is puzzled. She can’t remember how to figure this out. She knows that the area of a parallelogram is related to the area of a rectangle, but she can’t remember how to connect them.
Next, we can help Jillian figure out the area of one of the parallelograms by using the formula that we learned in this lesson.
\begin{align*}A & = bh\\ A & = 3.5(2)\\ A & = 7 \ square \ inches\end{align*}
Each parallelogram will be seven square inches.
Now we need 8 parallelograms. Let’s multiply our result by 8.
7 \begin{align*}\times\end{align*} 8 \begin{align*}=\end{align*} 56 square inches
Jillian will need 56 square inches of fabric. If we convert that to feet, 1 foot = 12 inches, so 1 foot x 1 foot = 12 inches x 12 inches = 144 square inches in every square foot. Jillian will need between 1/3 and 1/2 square feet of fabric to have enough for the 8 parallelograms.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Area
- the space within the perimeter of a figure or place. Area often refers to the surface or covering, the middle of a figure. Area is measured in square units.
- Parallelogram
- a quadrilateral with two pairs of opposite congruent sides.
- Rectangle
- a parallelogram with two pairs of opposite congruent sides and four 90 degree angles.
Technology Integration
Khan Academy, Area of a Parallelogram
Time to Practice
Directions: Find the area of each parallelogram using the given dimensions.
1. Base = 7 inches
Height = 4 inches
2. Base = 8 cm
Height = 2 cm
3. Base = 9 inches
Height = 4 inches
4. Base = 10 feet
Height = 5 feet
5. Base = 7 inches
Height = 6 inches
6. Base = 10 meters
Height = 8 meters
7. Base = 11 feet
Height = 9 feet
8. Base = 12 meters
Height = 10 meters
9. Base = 5 inches
Height = 4.5 inches
10. Base = 8.5 feet
Height = 2.5 feet
11. Base = 9.5 feet
Height = 3 feet
12. Base = 6.5 feet
Height = 3.5 feet
13. Base = 9.5 cm
Height = 2 cm
14. Base = 15 feet
Height = 12 feet
15. Base = 150 miles
Height = 20 miles
Directions: Use the given area and other dimension to find the missing base or height.
16. Area = 22 sq. inches
Base = 11 inches
17. Area = 50 sq. miles
Base = 10 miles
18. Area = 48 sq. inches
Base = 8 inches
19. Area = 30 sq. meters
Base = 15 meters
20. Area = 45 sq. feet
Height = 3 feet
21. Area = 88 sq. feet
Height = 8 feet
22. Area = 121 sq. feet
Height = 11 feet
23. Area = 160 sq. miles
Height = 20 miles
24. Area = 90 sq. meters
Height = 30 meters
25. Area = 100 sq. feet
Base = 25 feet
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