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.

You will put together all you have learned previously in this activity.
Image courtesy of www.imagesgoogle.com



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 units2 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 units2 and a perimeter of 14 units, the rectangle must be 4 units long and 3 units wide.


 

Try This:


This problem asks you to construct a rectangular garden.

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 units2 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.