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Area of Squares and Rectangles

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Area of Squares and Rectangles

Remember how Alex figured out the perimeter of the garden plot in the last Concept? What about the space inside the fence where Tania and Alex will plant?

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.

Alex needs to know how much area they will actually have to plant on. To figure this out, Alex needs the area of each garden plot.

This Concept will teach you all about area. Then you will be able to figure out the area of each plot with Alex.

Guidance

In the last Concept, you learned that the perimeter is the distance around the edge of a figure.

What about the space inside the figure?

We call this space the area of the figure. The area of a figure can also be called the surface of the figure. When we talk about carpeting or flooring or grass or anything that covers the space inside of a figure, we are talking about the area of that figure.

We can calculate the area of different shapes.

How can we figure out the area of a square?

To figure out the area of a square, we need to calculate how much space there is inside the square.

We can use a formula to help us with this calculation.

A=s \cdot s

In this formula, the little dot means multiplication. To figure out the area of the square we multiply one side times another side.

A=6 \ ft \cdot 6 \ ft

Here is what the problem looks like. Next, we multiply.

A &= 6 \cdot 6\\A &= ft \cdot ft

Here we are multiplying two different things. We multiply the actual measurement 6 \times 6 and we multiply the unit of measurement too, feet \times feet.

A &= 6 \times 6 = 36\\A &= ft \times ft = sq.ft \ \text{or} \ ft^2

Think about the work that we did before with exponents. When we multiply the unit of measurement, we use an exponent to show that we multiplied two of the same units of measurement together.

Take a minute and copy this formula for finding the area of a square into your notebook.

How can we find the area of a rectangle?

To find the area of a rectangle, we are going to use the measurements for length and width.

Here we have a rectangle with a length of 5 meters and a width of 3 meters. Just like the square, we are going to multiply to find the area of the rectangle. Here is our formula.

A=lw

To find the area of a rectangle, we multiply the length by the width.

A &= (5m)(3m)\\A &= 5 \times 3\\A &= meters \times meters

Here we have 5 meters times 3 meters. We multiply the measurement part 5 \times 3, then we multiply the units of measure.

Our final answer is 15 sq.m or 15 \ m^2

We can also use square meters or \ meters^2 to represent the unit of measure. When working with area, we must ALWAYS include the unit of measure squared. This helps us to remember that the units cover an entire area.

Take a minute to copy down the formula for finding the area of a rectangle into your notebook

Now let's practice.

Example A

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

Solution: 49 square inches

Example B

Find the area of a rectangle with a length of 12 cm and a width of 3 cm.

Solution: 36 square centimeters

Example C

Find the area of a square with a side length of 11 meters.

Solution: 121 square meters

Now back to Alex and the garden plot.

Alex has the dimensions of his garden plot, so now he can figure out the area. He will figure out the area of the square plot and then add that to the area of the rectangular plot. This will give him the total area of the garden.

The square plot has a side length of 9 feet.

A &= s \times s\\A &= 9 \times 9 = 81 \ sq. \ feet

The square plot has an area of 81 square feet.

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

A &= (12ft)(8ft)\\A &= 12 \times 8\\A &= feet \times feet

The rectangular plot has an area of 96 square feet.

Now he can add the two areas together.

81 + 96 = 177 square feet

This is our answer.

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
Area
the space inside the edges of a figure
Dimensions
the measurements that define a figure

Guided Practice

Here is one for you to try on your own.

In the last Concept, Alex discovered that if he put the square plot next to the rectangular plot that he wouldn't need as much fencing. Putting the plots together changed the perimeter of the plot.

Does it also change the area? Why or why not?

Answer

The area is the measurement of the space inside the perimeter. Therefore, the shape of the plot didn't change, so the area of the plot didn't change either. Therefore, the area of the two plots would not change if they were put next to each other.

Video Review

Khan Academy Area and Perimeter

James Sousa Area and Perimeter

James Sousa An Example of Area and Perimeter

Practice

Directions: Find the area of each of the following figures. Be sure to label your answer correctly.

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 13 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.

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