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# Lesson 16 — Activity 3: Constructing Rectangles When Given Both the Perimeter and the Area

# Lesson 16 — Activity 3: Constructing Rectangles when given both the Perimeter and the Area

#### Getting Ready

####
In activity 1, you learned how to construct a rectangle when given the perimeter.

#### In activity 2, you learned how to figure out the different lengths and widths of rectangles when given the area of the rectangle.

#### In this activity, you will learn how to construct a rectangle when you are given both the area and the perimeter.

#### There are two steps to constructing a rectangle when you know the area and the perimeter.

####
Step 1: Find all of the possibilities for the lengths and widths based on the given area.

Step 2: Use the lengths and widths that you found in Step 1 to find the proper length and width to meet the desired perimeter.

#### For example, let's say you are given the following information about a rectangle:

####
It has an area of 12 units^{2} and a perimeter of 14 units.

#### Step 1: Find all of the possibilities for the lengths and widths of the rectangle.

####
Using a width of 1 unit:

####

l = A ÷ w

l = 12 ÷ 1

l = 12 units

####

The rectangle could be 12 units long and 1 unit wide.

####
Using a width of 2 units:

####

l = A ÷ w

l = 12 ÷ 2

l = 6 units

####

The rectangle could be 6 units long and 2 units wide.

#### Using a width of 3 units:

####

l = A ÷ w

l = 12 ÷ 3

l = 4 units

####

The rectangle could be 4 units long and 3 units wide.

####
Step 2. Add up all of the lengths and widths to find the correct perimeter.

####
P = 12 + 12 + 1 + 1

P = 26 units

P = 6 + 6 + 2 + 2

P = 16 units

P = 4 + 4 + 3 + 3

P = 14 units (This is the correct perimeter.)

Based on the fact that the rectangle has to have an area of 12 units^{2} and a perimeter of 14 units, the rectangle must be 4 units long and 3 units wide.

P = 26 units

P = 6 + 6 + 2 + 2

P = 16 units

P = 4 + 4 + 3 + 3

P = 14 units (This is the correct perimeter.)

Based on the fact that the rectangle has to have an area of 12 units

^{2}and a perimeter of 14 units, the rectangle must be 4 units long and 3 units wide.

#### Try This:

Image courtesy of www.imagesgoogle.com

#### You have been asked to help design a rectangular garden in your neighbourhood.

Follow the steps above to construct a rectangular garden that has an area of 20 units^{2} and a perimeter of 18 units.

**Step 1:**

Using a width of 1 unit:

l = 20 ÷ 1

l = 20 units

Using a width of 2 units:

l = 20 ÷ 2

l = 10 units

Using a width of 3 units:

l = 20 ÷ 3

l = 6.666 (Try another one as this is not a whole number.)

Using a width of 4 units:

l = 20 ÷ 4

l = 5 units

**Step 2**:

P = 20 + 20 + 1 + 1

P = 42 units

P = 10 + 10 + 2 + 2

P = 24 units

P = 5 + 5 + 4 + 4

P = 18 units (This is the perimeter you were looking for.)

Therefore, the rectangle must be 5 units long and 4 units wide.

l = 20 ÷ 1

l = 20 units

Using a width of 2 units:

l = 20 ÷ 2

l = 10 units

Using a width of 3 units:

l = 20 ÷ 3

l = 6.666 (Try another one as this is not a whole number.)

Using a width of 4 units:

l = 20 ÷ 4

l = 5 units

P = 42 units

P = 10 + 10 + 2 + 2

P = 24 units

P = 5 + 5 + 4 + 4

P = 18 units (This is the perimeter you were looking for.)