# Areas of Rectangles and Parallelograms

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Areas of Rectangles and Parallelograms
Section 8.1

centimeters square centimeters Etc.
You measure the area of a figure by counting the number of square units that you can arrange to fill the figure completely. If sides are measured in use inches square inches feet square feet centimeters square centimeters Etc.

Do Now Each of these shapes is filled with squares that measure 1 foot by 1 foot. Find the total number of squares contained in each shape.

91 square feet 28 square feet 101 square feet 200 square feet 82 square feet 32 square feet

You probably already know many area formulas
You probably already know many area formulas. Think of the investigations in this chapter as physical demonstrations of the formulas that will help you understand and remember them. How can you find the area of these three rectangles?

The BASE tells you the number of squares on the bottom row
Any side of a rectangle can be called the base. The other side is called the height. The HEIGHT tells you the number of rows of squares How can you determine the total number of squares in a rectangle? Base = 3 units or there are 3 squares on the bottom row Base = 5 units or there are 5 squares on the bottom row Height = 5 units or there are 5 rows of squares Height = 3 units or there are 3 rows of squares

A= bh Rectangle Area Conjecture
If a rectangle has a base equal to b and a height equal to h, then its area A can be found by A= bh The base tells you how many squares are on the bottom row. h The height tells you how many rows of squares are in the rectangle. h b b

20 25

The area formula for rectangles can help you find the areas of many other shapes.
You know how to find this area. How can you find the area of this green triangle? Explain your reasoning.

Example A Find the area of this shape.

Example A Think of the shape this way
The area is or 52 square units.

You can also use the area formula for a rectangle to find the area formula for a parallelogram.
Just as with a rectangle, any side of the parallelogram can be called the base. BUT THE HEIGHT OF THE PARALLELOGRAM IS NOT NECESSARILY THE LENGTH OF A SIDE. base base

An altitude is any segment from one side of the parallelogram perpendicular to a line through the opposite side. The length of the altitude is the height.

The altitude can be inside or outside the parallelogram
The altitude can be inside or outside the parallelogram. No matter where you draw the altitude to the base, its height should be the same, because the opposite sides are parallel.

Area Formula for Parallelograms
You will received a cutout of a parallelogram. Select a side for the base and label it. Draw a line that represents the height of the parallelogram. How long is your base? How tall is your height? height base

height base Cut the parallelogram along the height to form two pieces. What is the name for these pieces? Rearrange the pieces to form another familiar shape. Do you have the correct measurements to find the area of this new shape? Explain how you can find its area. The slanted line is a side of the parallelogram. Is the slanted line longer or shorter than the height? Do you think its length could help you find the area of a parallelogram?

Do the number of rows change as you change the figure?
height base Slide the shape back to the parallelogram. Which measurements would you use to find the area of the parallelogram? What do these measurements describe about the parallelogram? Do the number of squares in the bottom row change as you change the figure from a parallelogram to a rectangle? Do the number of rows change as you change the figure? base height

The figure is a parallelogram
The figure is a parallelogram. Its base is b, its height is h, and the other pair of sides measure s, which is more than h. The area of this parallelogram, or the number of squares contained in the paralleogram can be found by _______________________________________ A formula would be A = bh

Example B Find the height of a parallelogram that has area 7.13 square meters and base length 2.3 meters.

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