2.2: Perimeter and Area
Introduction
The Garden Plot
While Tania has been working on her tomato plants, Alex has been working on designing the garden plot. He knows that he wants two plots, one to be in the shape of a square and one to be the shape of a rectangle. His square plot has a length and width of 9 feet.
His rectangle plot has a length of 12 feet and a width of 8 feet.
Tania and Alex live near some woods and they have seen deer and rabbits in their back yard on several different occasions. Because of this, Alex knows that he will need to put some fencing around both of the garden plots. He is puzzled about how much fencing he will need. Alex needs to know the perimeter (the distance around the border) of each plot. Next, he needs to know how much area they will actually have to plant on. To figure this out, Alex needs the area of each garden plot. Alex has another idea too. He wonders what the dimensions will be if he puts the square plot right up against the rectanglar plot. Will this change the amount of fencing he will need? Will this change the area of the garden plot?
He has drawn some sketches of his garden design, but can’t seem to figure out the dimensions. Alex is having a very tough time. He can’t remember how to calculate these two important measurements.
In this lesson, you will learn all about perimeter and area in order to help Alex with his garden plan. Pay close attention so that you can help Alex to figure out the measurements needed for the vegetable garden.
What You Will Learn
In this lesson, you will learn all that you need to know to help Alex with his garden plan.
You will learn the following skills:
- Finding the perimeters of squares and rectangles using formulas
- Finding the areas of squares and rectangles using formulas
- Solving for unknown dimensions using formulas when given the perimeter or the area
- Solving real-world problems involving perimeter and area, including irregular figures made of rectangles and squares.
Teaching Time
I. Finding the Perimeter of Squares and Rectangles
What do we mean when we use the word perimeter?
The perimeter is the distance around the edge of an object. We can find the perimeter of any figure. When working on a word problem, there are some key words that let us know that we will be finding the perimeter of a figure. Those key words are words like edges, fencing and trim to name a few.
We can find the perimeter of squares and rectangles.
Look at a square and see how we can figure out the distance around the square.
Here is a square. Notice that we have only one side with a given measurement. The length of one side of the square is 5 feet.
Why is that? Why is there only one side with a measurement on it?
Think about the definition of a square. A square has four congruent sides. That means that the sides of a square are the same length. Therefore, we only need one side measurement and we can figure out the measurement around the other three edges of the square.
How can we use this information to figure out the perimeter of the square?
We can figure out the perimeter of the square by simply adding the lengths of each of the sides.
In this example, we would add 5 + 5 + 5 + 5 = 20 feet. This is the perimeter of this square.
We can use a formula to give us a shortcut to finding the perimeter of a square. A formula is a way of solving a particular problem.
When figuring out the perimeter of a square, we can use this formula to help us.
\begin{align*}& P=4s\\ & or\\ & P=s+s+s+s\end{align*}
The \begin{align*}P\end{align*} in the formula stands for perimeter.
The \begin{align*}s\end{align*} stands for the measure of the side.
Notice that in the first version of the formula we can take four and multiply it by the length of the side. Remember that multiplication is a shortcut for repeated addition.
The second formula shows us the repeated addition.
Either formula will work.
Now that you are in grade 6, it is time for you to begin using formulas.
Let’s apply this formula to the square that we looked at with 5 ft on one side.
\begin{align*}P &= s+s+s+s\\ P &= 5+5+5+5\\ P &= 20 \ ft\end{align*}
We can also use the formula with multiplication to get the same answer.
\begin{align*}P &= 4s\\ P &= 4(5)\\ P &= 20 \ ft\end{align*}
Take a minute and copy these two formulas into your notebook.
How can we use this information to find the perimeter of a rectangle?
First, let’s think about the definition of a rectangle.
A rectangle has opposite sides that are congruent. In other words, the two lengths of the rectangle are the same length and the two widths of a rectangle are the same width.
Let’s look at a diagram of a rectangle.
Notice that the side lengths have “ next to them, this means inches. When used this way, the symbol means inches.
When we figure out the perimeter of the rectangle, we can’t use the same formula that we did when finding the perimeter of the square.
Why is this?
A square has four sides of equal length. A rectangle has two equal lengths and two equal widths.
Here is our formula for finding the perimeter of a rectangle.
\begin{align*}P=2l+2w\end{align*}
Since we have two lengths that have the same measure and two widths that have the same measure, we can add two times one measure and two times the other measure and that will give us the distance around the rectangle.
Now let’s apply this to our example.
If we have a rectangle with a length of 8 inches and a width of 6 inches, we can substitute these measures into our formula and solve for the perimeter of the rectangle.
\begin{align*}P &= 2l+2w\\ P &= 2(8)+2(6)\\ P &= 16+12\\ P &= 28 \ inches\end{align*}
Take a minute and copy the formula for finding the perimeter of a rectangle into your notebook.
Here are a few for you to try on your own. Be sure to label your answer with the correct unit of measurement.
- Find the perimeter of a square with a side length of 7 inches.
- Find the perimeter of a rectangle with a length of 9 feet and a width of 3 feet.
- Find the perimeter of a square with a side length of 2 centimeters.
Take a few minutes to check your work with a friend.
II. Finding the Area of Squares and Rectangles
We just finished learning about perimeter, the distance around the edge of a figure.
What about the space inside the figure?
We call this space the area of the figure. The area of a figure can also be called the surface of the figure. When we talk about carpeting or flooring or grass or anything that covers the space inside of a figure, we are talking about the area of that figure.
We can calculate the area of different shapes.
How can we figure out the area of a square?
Let’s look at an example to help us.
To figure out the area of a square, we need to calculate how much space there is inside the square.
We can use a formula to help us with this calculation.
\begin{align*}A=s \cdot s\end{align*}
In this formula, the little dot means multiplication. To figure out the area of the square we multiply one side times the side.
\begin{align*}A=6 \ ft \cdot 6 \ ft\end{align*}
Here is what the problem looks like. Next, we multiply.
\begin{align*}A &= 6 \cdot 6\\ A &= ft \cdot ft\end{align*}
Here we are multiplying two different things. We multiply the actual measurement 6 \begin{align*}\times\end{align*} 6 and we multiply the unit of measurement too, feet \begin{align*}\times\end{align*} feet.
\begin{align*}A &= 6 \times 6 = 36\\ A &= ft \times ft = sq.ft \ \text{or} \ ft^2\end{align*}
Think about the work that we did before with exponents. When we multiply the unit of measurement, we use an exponent to show that we multiplied two of the same units of measurement together.
Take a minute and copy this formula for finding the area of a square into your notebook.
How can we find the area of a rectangle?
To find the area of a rectangle, we are going to use the measurements for length and width.
Let’s look at an example and then figure out the area of the rectangle using a new formula.
Here we have a rectangle with a length of 5 meters and a width of 3 meters.
Just like the square, we are going to multiply to find the area of the rectangle.
Here is our formula.
\begin{align*}A=lw\end{align*}
To find the area of a rectangle, we multiply the length by the width.
\begin{align*}A &= (5m)(3m)\\ A &= 5 \times 3\\ A &= meters \times meters\end{align*}
Here we have 5 meters times 3 meters.
We multiply the measurement part 5 \begin{align*}\times\end{align*} 3, then we multiply the units of measure.
Our final answer is 15 sq.m or \begin{align*}15 \ m^2\end{align*}
We can also use square meters or meters\begin{align*}^2\end{align*} to represent the unit of measure. When working with area, we must ALWAYS include the unit of measure squared. This helps us to remember that the units cover an entire area.
Take a minute to copy down the formula for finding the area of a rectangle into your notebook
Here are a few for you to try on your own. Be sure to include the unit of measurement in your answer.
- Find the area of a square with a side length of 7 inches.
- Find the area of a rectangle with a length of 12 cm and a width of 3 cm.
- Find the area of a square with a side length of 11 meters.
Take a minute and check your work with a peer.
III. Solving for Unknown Dimensions Using Formulas
The side length of a square or the length and width of a rectangle can be called the dimensions or the measurements of the figure.
We just finished figuring out the area and perimeter of squares and rectangles when we were given the dimensions of the figure. Can we do this work backwards?
Can we figure out the dimensions of a square when we have been given the perimeter or area of the square?
Hmmmm. This is a bit tricky. We will still need to use the formula, but we will need to “think backwards” in a way. Let’s look at an example and see how this works out. We’ll start by figuring out the dimensions of a square when given the perimeter of the square.
Example
If the perimeter of the square is 12 inches, what is the side length of the square?
To complete this problem, we are going to need to work backwards. Let’s start by using the formula for the perimeter of a square.
\begin{align*}P=4s\end{align*}
Next, we fill in the information that we know. We know the perimeter or \begin{align*}P\end{align*}.
\begin{align*}12=4s\end{align*}
We can ask ourselves, “What number times four will give us 12?”
The answer is 3.
We can check our work by substituting 3 in for s to see if we have a true statement.
\begin{align*}12 &= 4(3)\\ 12 &= 12\end{align*}
Our answer checks out.
Now let’s look at how we can figure out the side length of a square when we have been given the area of the square.
Example
Area = 36 sq. in.
We know that the area of the square is 36 square inches. Let’s use the formula for finding the area of a square to help us.
\begin{align*}A &= s \times s\\ 36 &= s \times s\end{align*}
We can ask ourselves, “What number times itself will give us 36?”
Our answer is 6.
Because we have square inches, we know that our answer is 6 inches.
We can check our work by substituting 6 into the formula for finding the area of a square.
\begin{align*}36 &= 6 \times 6\\ 36 &= 36\end{align*}
Our answer checks out.
Here are few for you to try on your own.
- What is the side length of a square that has a perimeter of 48 feet?
- What is the side length of a square that has a perimeter of 56 feet?
- What is the side length of a square that has an area of 64 sq. inches?
- What is the side length of a square that has an area of 121 sq. miles?
Take a minute and check your work with a peer.
Did you remember to use the correct unit of measurement?
Real Life Example Completed
The Garden Plot
Now that you have learned all about area and perimeter, you are ready to help Alex figure out the questions he wants answered about his garden plot.
Let’s look at the problem once again.
While Tania has been working on her tomato plants, Alex has been working on designing the garden plot.
He knows that he wants two plots, one to be in the shape of a square and one to be the shape of a rectangle.
His square plot has a length and width of 9 feet.
His rectangle plot has a length of 12 feet and a width of 8 feet.
Tania and Alex live near some woods and they have seen deer and rabbits in their back yard on several occasions. Because of this, Alex knows that he will need to put some fencing around both of the garden plots.
He is puzzled about how much fencing he will need. Alex needs to know the perimeter (the distance around the border) of each plot.
Next, he needs to know how much area they will actually have to plant on. To figure this out, Alex needs the area of each garden plot. He wants to have the largest area to plant that he can.
Alex has another idea too. He wonders what the dimensions will be if he puts the square plot right up against the rectangle plot.
Will this change the amount of fencing he will need?
Will this change the area of the garden plot?
He has drawn some sketches of his garden design, but can’t seem to figure out the dimensions.
The first thing that we need to do is to underline all of the important information in the problem. That will include dimensions and other pertinent information that we need to look at to help Alex find the answers to his questions. The underlining has been done for you.
There are two main designs that Alex is working with.
- A square plot and a rectangle plot separate.
- A square plot and a rectangle plot put together.
Let’s start by looking at each garden plot separate.
First, we find the perimeter and area of the square plot.
The square plot is 9 feet long on one side.
\begin{align*}P &= 4s\\ P &= 4(9) = 36 \ feet\end{align*}
The square plot has a perimeter of 36 feet. He will need 36 feet of fencing for the small plot.
\begin{align*}A &= s \times s\\ A &= 9 \times 9 = 81 \ sq. \ feet\end{align*}
The square plot has an area of 81 square feet.
Now let’s find the perimeter and the area of the rectangle plot.
The rectangle plot has a length of 12 feet and a width of 8 feet.
\begin{align*}& P = 2l + 2w\\ & P = 2(12) + 2(8)\\ & P = 24 + 16 = 40 \ \text{feet of fencing is needed for the rectangle plot.}\\ & A = l \times w\\ & A = 12 \times 8 = 96 \ sq. \ feet\\ &\text{The rectangle plot has an area of} \ 96 \ sq. \ feet.\end{align*}
Now we know the perimeter and area of each garden plot if Alex chooses to keep them separate.
What happens if he puts them together?
If Alex puts the square plot right next to the rectangle plot, then he will have a plot that is an irregular shape.
Let’s look at a diagram of what this will look like.
The first thing to notice is that the area of the figure has not changed.
We were able to add the area of the square and the area of the rectangle and here is the area of the entire irregular garden.
\begin{align*}A = 81 + 96 = 177 \ sq. feet.\end{align*}
The amount of fencing, however, has changed. One side of each figure has almost completely disappeared. How does this affect the perimeter of the figure? 3 sides of the rectangle now = 12 + 12 + 8 = 32 feet 4 sides of the square \begin{align*}=\end{align*} 9 \begin{align*}\times\end{align*} 3 \begin{align*}=\end{align*} 27 + 1 = 28 feet Notice that we don’t count the 1 ft twice. It overlaps both figures. We counted it in the rectangle, so we don’t need to count it in the square. Next, we can add the two perimeters together. This will give us the distance around the entire irregular figure. 32 + 28 = 60 feet of fencing is needed for the irregular garden plot. Alex takes a look at all of the work that we have done. Because he will need less fencing, Alex decides to put the two plots together to make one large irregular plot. For separate plots, Alex would have needed 76 feet of fencing. For the irregular plot, Alex will only need 60 feet of fencing. With the money he is saving, Alex figures that he and Tania can buy more seeds.
Vocabulary
Here are the vocabulary words that were used throughout this lesson.
- Perimeter
- the distance around the edge of a figure.
- Square
- a figure with four congruent sides
- Formula
- a way or method of solving a problem
- Rectangle
- a figure that has opposite sides that are congruent
- Area
- the space inside the edges of a figure
- Dimensions
- the measurements that define a figure
Technology Integration
Khan Academy Area and Perimeter
James Sousa Area and Perimeter
James Sousa An Example of Area and Perimeter
You will need to register with this website. Here is a video performance of a teacher with her perimeter rap. A fun way to remember how to find the perimeter of a figure!
http://www.teachertube.com/viewVideo.php?video_id=157&title=Mrs__Burk_Perimeter_Rap&ref –
- http://www.teachertube.com/viewVideo.php?video_id=157&title=Mrs__Burk_Perimeter_Rap&ref – Here is a video performance of a teacher with her perimeter rap. A fun way to remember how to find the perimeter of a figure!
Time to Practice
Directions: Find the perimeter of each of the following squares and rectangles.
1. A square with a side length of 6 inches.
2. A square with a side length of 4 inches.
3. A square with a side length of 8 centimeters.
4. A square with a side length of 12 centimeters.
5. A square with a side length of 9 meters.
6. A rectangle with a length of 6 inches and a width of 4 inches.
7. A rectangle with a length of 9 meters and a width of 3 meters.
8. A rectangle with a length of 4 meters and a width of 2 meters.
9. A rectangle with a length of 17 feet and a width of 12 feet.
10. A rectangle with a length of 22 feet and a width of 18 feet.
Directions: Find the area of each of the following figures. Be sure to label your answer correctly.
11. A square with a side length of 6 inches.
12. A square with a side length of 5 centimeters.
13. A square with a side length of 7 feet.
14. A square with a side length of 8 meters
15. A square with a side length of 12 meters.
16. A rectangle with a length of 6 meters and a width of 3 meters.
17. A rectangle with a length of 5 meters and a width of 2 meters.
18. A rectangle with a length of 11 feet and a width of 12 feet.
19. A rectangle with a length of 9 meters and a width of 22 meters.
20. A rectangle with a length of 11 feet and a width of 19 feet.
Directions: Find the side length of each square given its perimeter.
21. P = 24 inches
22. P = 36 inches
23. P = 50 inches
24. P = 88 centimeters
25. P = 90 meters
Directions: Find the side length of each square given its area.
26. A = 64 sq. inches
27. A = 49 sq. inches
28. A = 121 sq. feet
29. A = 144 sq. meters
30. A = 169 sq. miles