10.5 Area of Parallelograms
Learning Objectives
- I can recognize the area of a parallelogram as the area of a related rectangle.
- I can find the area of parallelograms given base and height.
- I can find unknown dimensions of parallelograms given area and another dimension.
- I can estimate actual areas approximated by parallelograms in scale drawings.
- I can solve real world problems involving area of parallelograms.
Introduction
The Quilt Block
Jillian’s grandmother is coming to spend the summer in Brainerd 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 4.2 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.
Guided Learning
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 base and a height. We can find the area of a rectangle by multiplying the base times the height.
5 10 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 base of 9 and the height of 2 and get the area of 18 square units.
Next, we look at a parallelogram.
Remember learning to find the area of irregular shapes? If we take off the two yellow triangles on the ends, you can see that the parallelogram is a lot like the rectangle. The yellow triangles will combine to form more whole units. There are 16 whole square units within this figure. We can also calculate the area of parallelograms using the base and height of the shape.
In this parallelogram we have a base of 8 and a height of 2.
8 2 16
The area of the parallelogram is 16 square units.
How else could we do that?
Use the formula for Area of a Parallelogram: A = bh
A = bh = 8(2) = 16 square units
Take a moment to write the formula for area of a parallelogram in your notes.
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 A
To find the area of this parallelogram, we multiply the base times the height.
Guided Practice
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!
Solutions
1. A = bh = 6(2) = 12 sq. cm
2. A = bh = 9(3.5) = 31.5 sq. in.
3. A = bh = 5(3) = 15 sq. in.
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 B
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 6 8
The measurement of the base is 8 inches.
We can also solve this by writing an equation with the area formula and solving for the missing variable.
48 = b(6)
48/6 = b(6)/6
8 = b
The same answer was found. 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 C
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 9 6
The measurement of the height is 6 feet.
Using the area formula and solving the equation, we should arrive with the same solution.
54 = 9h
54/9 = 9h/9
6 = h
The height is 9 feet.
Guided Practice
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?
Solutions
1. A = bh 25 = 5h h = 5 meters
2. A = bh 81 = 27h h = 3 feet
3. A = bh 36 = 2b b = 18 inches
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 ” 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 D
Let’s say that we wanted to use this map and the parallelogram to estimate the area in and around Berlin. We will use the parallelogram to outline the area around Berlin we wish to calculate.
Using estimation, we could say that the base of the parallelogram is ”. Using the scale, 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 16 384 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 in Brainerd 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 4.2 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.
A = 8.4 square inches
Each parallelogram will be 8.4 square inches.
Now we need 8 parallelograms. Let’s multiply our result by 8.
8.4 8 67.2 square inches
Jillian will need 67.2 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 about 1/2 square feet of fabric to have enough for the 8 parallelograms.
Review
- Use the formula to find the area of parallelograms. Area = base x height
Area
Area is the space inside the edges of a figure.
Parallelogram
Quadrilateral with opposite sides congruent and parallel is called a parallelogram.
Rectangle
A four sided figure with opposite sides that are congruent and with four right angles is called a rectangle.