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Perimeter of Squares and Rectangles

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Have you ever put up a fence? Did the fence go around the edge of a plot of land? Was the shape of the land square or rectangular? If you have ever done this, then you have measured perimeter.

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.

Use this Concept to solve this dilemma.

Guidance

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.

Let's learn how to 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 case, 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.

$& P=4s\\& or\\& P=s+s+s+s$

The $P$ in the formula stands for perimeter. The $s$ 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.

$P &= s+s+s+s\\P &= 5+5+5+5\\P &= 20 \ ft$

We can also use the formula with multiplication to get the same answer.

$P &= 4s\\P &= 4(5)\\P &= 20 \ ft$

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. 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.

$P=2l+2w$

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. 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.

$P &= 2l+2w\\P &= 2(8)+2(6)\\P &= 16+12\\P &= 28 \ inches$

Take a minute and copy the formula for finding the perimeter of a rectangle into your notebook.

Now let's practice.

Example A

Find the perimeter of a square with a side length of 7 inches.

Solution: 28 inches

Example B

Find the perimeter of a rectangle with a length of 9 feet and a width of 3 feet.

Solution: 24 feet

Example C

Find the perimeter of a square with a side length of 2 centimeters.

Solution: 8 centimeters

Now back to Alex and the garden plot. Have you figured out what Alex should do? Here is the original problem once again.

Alex is trying to figure out the perimeter of a square plot, a rectangular plot and the perimeter of a plot where the square and the rectangle are next to each other. Let's start with the square plot.

$P &= 4s\\P &= 4(9) = 36 \ feet$

The square plot has a perimeter of 36 feet. He will need 36 feet of fencing for the small plot.

The rectangular plot has a length of 12 feet and a width of 8 feet.

$P &= 2l+2w\\P &= 2(12)+2(8)\\P &= 24+16\\P &= 40 \ feet$

Alex will need 40 feet of fencing for the rectangular plot.

Vocabulary

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

Guided Practice

Here is one for you to try on your own.

What would happen is Alex put the two plots together? Would he need more fencing or less?

If Alex put the square plot next to the rectangular plot, then one side of the square plot would not be needed and almost one side of the rectangular plot would not be needed.

We can work with the three sides of the square plot and the three sides of the rectangular plot first.

The square plot has three sides that are each 9 feet long. Therefore, Alex will need 27 feet of fencing for those three sides of the square plot.

The rectangular plot has one side that is 8 feet wide and two sides that are twelve feet wide. Alex will need 32 feet of fencing for the three sides of the rectangular plot.

The combined side will only need one side of fencing because the length of the square plot is 9 feet, but the width of the rectangle plot is 8 feet, leaving only one foot to fence.

Here is how we can calculate perimeter.

$P = 27 + 32 + 1 = 60$

Alex will only need 60 feet of fencing if he combines both plots.

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.

11. A square with a side length of 16 feet.

12. A square with a side length of 18 feet.

13. A square with a side length of 21 feet.

14. A rectangle with a length of 18 feet and a width of 13 feet.

15. A rectangle with a length of 60 feet and a width of 27 feet.

16. A rectangle with a length of 57 feet and a width of 22 feet.