Jennifer has carpet in her square bedroom. She decides to also purchase carpet for her living room which is rectangular in shape and 9 feet longer than her bedroom. The area of the carpet required in the living room is given by the quadratic expression below, where x represents the side length, in feet, of the carpet in the bedroom. Match each part of the expression with what it represents. Tiles:

1 the width of the carpet in the living room

2 the length of the carpet in the living room

3 the area of the carpet in the bedroom

4 the increase in the area of carpet needed for the living room

Pairs:

1 the second-degree term of the expression x2 + 9x

2 the monomial, x, a factor of the expression x2 + 9x

3 the first-degree term of the expression x2 + 9x

4 the binomial, (x + 9), a factor of the expression x2 + 9x

Sometimes, you will be given either the area or the perimeter in a problem and you will be asked to calculate the value you are not given. For example, you may be given the perimeter and be asked to calculate area; or, you may be given the area and be asked to calculate the perimeter. Let’s go through a few examples of what this would look like:

Area and Perimeter Example 1

Valery has a large, square room that she wants to have carpeted. She knows that the perimeter of the room is 100 feet, but the carpet company wants to know the area. She knows that she can use the perimeter to calculate the area.

What is the area of her room?

We know that all four sides of a square are equal. Therefore, in order to find the length of each side, we would divide the perimeter by 4. We would do this because we know a square has four sides, and they are each the same length and we want the division to be equal. So, we do our division-100 divided by 4-and get 25 as our answer. 25 is the length of each side of the room. Now, we just have to figure out the area. We know that the area of a square is length times width, and since all sides of a square are the same, we would multiply 25 x 25, which is 625. Thus, she would be carpeting 625 square feet.

Area and Perimeter Example 2

Now let’s see how we would work with area to figure out perimeter. Let’s say that John has a square sandbox with an area of 100 square feet. He wants to put a short fence around his sandbox, but in order to figure out how much fence material he should buy, he needs to know the perimeter. He knows that he can figure out the perimeter by using the area.

What is the perimeter of his sandbox?

We know that the area of a square is length times width. In the case of squares, these two numbers are the same. Therefore, we need to think, what number times itself gives us 100? We know that 10 x 10 = 100, so we know that 10 is the length of one side of the sandbox. Now, we just need to find the perimeter. We know that perimeter is calculated by adding together the lengths of all the sides. Therefore, we have 10 + 10 + 10 + 10 = 40 (or, 10 x 4 = 40), so we know that our perimeter is 40 ft. John would need to buy 40 feet of fencing material to make it all the way around his garden.

Calculating Area and Perimeter Using Algebraic Equations

So far, we have been calculating area and perimeter after having been given the length and the width of a square or rectangle. Sometimes, however, you will be given the total perimeter, and a ratio of one side to the other, and be expected to set up an algebraic equation (using variables) in order to solve the problem. We’ll show you how to set this up so that you can be successful in solving these types of problems.

Eleanor has a room that is not square. The length of the room is five feet more than the width of the room. The total perimeter of the room is 50 ft. Eleanor wants to tile the floor of the room. How many square feet (ft 2) will she be tiling?

In this problem, we will be calculating area, but first we’re going to use the perimeter to figure out the length and width of the room.

First, we have to assign variables to each side of the rectangle. X is the most often used variable, but you can pick any letter of the alphabet that you’d like to use. For now, we’ll just keep things simple and use x. To assign a variable to a side, you first need to figure out which side they give you the least information about. In this problem, it says the length is five feet longer than the width. That means that you have no information about the width, but you do have information about the length based on the width. Therefore, you’re going to call the width (the side with the least information) x. Now, the width = x, and x simply stands for a number you don’t know yet. Now, you can assign a variable to the length. We can’t call the length x, because we already named the width x, and we know that these two measurements are not equal. However, the problem said that the length is five feet longer than the width. Therefore, whatever the width (x) is, we need to add 5 to get the length. So, we’re going to call the length x + 5.