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# Area

Area is the measure of the amount of surface covered by something. Area formulas for different shapes are sometimes different, but for the most part, area is calculated by multiplying length times width. This is used when calculating area of squares and rectangles. Once you have the number answer to the problem, you need to figure out the units. When calculating area, you will take the units given in the problem (feet, yards, etc) and square them, so your unit measure would be in square feet (ft.2) (or whatever measure they gave you).

### Area Example 1

Let’s try an example. Nancy has a vegetable garden that is 6 feet long and 4 feet wide. It looks like this:

Nancy wants to cover the ground with fresh dirt. How many square feet of dirt would she need?

We know that an answer in square feet would require us to calculate the area. In order to calculate the area of a rectangle, we multiply the length times the width. So, we have 6 x 4, which is 24. Therefore, the area (and amount of dirt Nancy would need) is 24 square feet.

### Area Example 2

Let’s try that one more time. Zachary has a wall that he would like to paint. The wall is 10 feet wide and 16 feet long. It looks like this:

How many square feet will he be painting?
We know that, in order to solve for square feet, we need to find the area. In order to find area of this rectangle, we would multiply the length times the width. When you multiply 10 x 16, you get 160. Therefore, your answer would be that Zachary will be painting 160 square feet of wall.
160

## Using Area and Perimeter Together

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.

Now that we’ve named each side, we can say that width = x, and length = x + 5. Here’s a picture of what this would look like:

Next, we need to set up an equation using these variables and the perimeter in order to figure out the length of each side. Remember, when calculating perimeter you add all four sides together. Our equation is going to look the same way, just with x’s instead of numbers. So, our equation looks like this:

x + x + x + 5 + x + 5 = 50

Now, we need to make this look more like an equation we can solve. Our first step is to combine like terms, which simply means to add all the x’s together, and then add the whole numbers together (for more help on this, see Combining Like Terms).

Once we combine like terms, our equation looks like this:

4x + 10 = 50

Next, we follow the steps for solving equations. (For additional help with this, see Solving Equations). We subtract 10 from each side of the equation, which leaves us with the following:

4x = 40

Now, we have to get x by itself, which means getting rid of the 4. In order to do this, we need to perform the opposite operation of what’s in the equation. So, since 4x means multiplication, we need to divide by 4 to get x alone. But remember, what we do to one side, we have to do to the other side. After dividing each side by 4, we get:

x = 10

Next, we have to interpret what this means. We look back and recall that we named the width x, so the width is 10. Now, we need to figure out the length. We named the length x + 5, so that means we have to substitute 10 in for x, and complete the addition. Therefore, we have 10 + 5, which gives us 15. So, our length is 15.

Now, we need to look back and remember that the problem asked us to calculate the area of the floor that Eleanor will be tiling. We know that in order to calculate area, we need to multiply the length times the width. We now have both the length and the width, so we simply set up a multiplication problem, like this: 10 x 15 = ? We multiply the two numbers together, and get 150.

## Area and Perimeter Practice Problems

Now, we’ll give you several practice problems so that you can try calculating area and perimeter on your own.

1. Leah has a flower garden that is 4 meters long and 2 meters wide. Leah would like to put bricks around the garden, but she needs to know the perimeter of the garden before she buys the bricks.

What is the perimeter of Leah’s garden (in meters)?
To find perimeter, we add together all 4 sides of the garden. We are told that the garden is 4 meters long and 2 meters wide. Therefore, we would add together 4 + 4 + 2 + 2 = 12 meters. So, the perimeter of the garden is 12 meters.
{12| 12 m| 12 meters| 12 meter}

2. David has a rug that is square, and the length of one side is 5 feet. He has an open floor space in his living room that is 36 square feet.

Would the rug fit perfectly, be too big, or be too small for the space he has? (Answer Choices: fit perfectly, too big, too small)
The rug would be too small. The area of the floor space is 36 square feet (the problem told us this). The area of the rug is only 25 square feet (we got this by multiplying 5 x 5 = 25). So, he could put the rug there, but it would be too small (it would not cover the entire area).
too small

3. Debbie has pool in her back yard that has a perimeter of 64 feet. The length of the pool is 2 feet longer than the width. Debbie wants to buy a cover for the pool, and needs to know how many square feet she needs to cover.

How many square feet (ft2) is Debbie’s pool? (hint: if you can, set up an algebraic equation to solve this problem!)
To solve this, we would set up an algebraic equation. We would call the width x, and the length x + 2, because the problem says that the length is 2 longer than the width. We know that to figure out perimeter, we need to add together all four sides. Thus, our equation would look like this: x + x + x + 2 + x + 2 = 64. This is shown in the diagram below:

When we combine like terms, we would get 4x + 4 = 64. Then, we would continue solving normally by subtracting 4 from each side, so the equation would simplify to 4x = 60. Lastly, we would divide each side by 4 (to get x by itself) and we would reach the conclusion x = 15. Thus, the width of the pool is 15 ft. However, the length is 2 more than the width, which means we would have to add 2 to the width; 2 + 15 = 17. Now, we have the length (17) and the width (15). The question asked for the area (square feet) of Debbie’s pool. To find area, you need to multiply the length times the width. Therefore, you would multiply 17 x 15, which gives us 255. Thus, Debbie would need a cover for her pool that is 255 feet2.
{255|255 sq ft|255 square feet|255 ft2|255 ft²|255 ft2}

4. Hector is planting a square garden in front of his house. He wants to plant carrots in the garden. He knows he can plant the carrots one foot apart. He has six feet across his yard (length) and he can plant carrots four feet deep (width).

How many square feet (ft2) does he have to plant carrots?
We know that area is calculated by multiplying the length times the width, so you would multiply 4 x 6 and get 24 square feet as the amount of space that he can plant in.
{24| 24 ft sq|24 sq ft|24 ft²}

5. Amanda is building a house, and she’s trying to calculate the area of her bedroom. She knows that the living room is 22 feet long and 20 feet wide. She was told that her bedroom should be half of the area of the living room.

What will the area of her bedroom be?
We know that the problem says the area of her bedroom will be 1/2 of the area of the living room. Thus, we first have to figure out the area of the living room. The dimensions of the living room are 22 feet by 20 feet, so we have to multiply these two numbers together to get the area. 22 x 20 = 440 square feet. Next, we have to divide that number by 2, because her bedroom is only half the area of that room. So, 440 divided by 2 equals 220. Therefore, the area of Amanda’s room would be 220 ft2.
{220| 220 ft sq| 220 square feet|220 ft²|220 sq ft}