Lesson 5Areas of Parallelograms

Learning Goal

Let’s investigate the area of parallelograms some more.

Learning Targets

I can identify pairs of base and height of a parallelogram.
I can use the area formula to find the area of any parallelogram.

Lesson Terms

base (of a parallelogram or triangle)
height (of a parallelogram or triangle)
parallelogram

Warm Up: A Parallelogram and Its Rectangles

Elena and Tyler were finding the area of this parallelogram:

Move the slider to see how Tyler did it:

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Move the slider to see how Elena did it:

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How are the two strategies for finding the area of a parallelogram the same? How they are different?

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Elena and Tyler were finding the area of this parallelogram:

Here is how Elena did it:

A right triangle is cut off the left side of the parallelogram and moved to the right side of the parallelogram to form a rectangle.

Here is how Tyler did it:

A cut perpendicular to a side of a parallelogram is made and the two pieces of the parallelogram are rearranged into a rectangle.

How are the two strategies for finding the area of a parallelogram the same? How they are different?

Activity 1: Finding the Formula for Area of Parallelograms

For each parallelogram:

Identify a base and a corresponding height, and record their lengths in the table that follows.
Find the area and record it in the right-most column.

In the last row, write an expression using $b$ and $h$ for the area of any parallelogram.

parallelogram	base (units)	height (units)	area (sq units)
$A$
$B$
$C$
$D$
any parallelogram	$b$	$h$

Are you ready for more?

Problem 1

What happens to the area of a parallelogram if the height doubles but the base is unchanged? If the height triples? If the height is 100 times the original?

Problem 2

What happens to the area if both the base and the height double? Both triple? Both are 100 times their original lengths?

Activity 2: More Areas of Parallelograms

Problem 1

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Calculate the area of the given figure in the applet. Then, check if your area calculation is correct by clicking the Show Area checkbox.
Uncheck the Area checkbox. Move one of the vertices of the parallelogram to create a new parallelogram. When you get a parallelogram that you like, sketch it and calculate the area. Then, check if your calculation is correct by using the Show Area button again.
Repeat this process. Draw and label the parallelogram with its measurements and the area you calculated.
Repeat this process one last time. Draw and label the parallelogram with its measurements and the area you calculated.

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Find the area of each parallelogram. Show your reasoning.

Problem 2

In Parallelogram b of the first problem, what is the corresponding height for the base that is 10 cm long? Explain or show your reasoning.

Problem 3

Explain why their areas are equal.
Drag points to create two new parallelograms that are not identical copies of each other but that have the same area as each other. Sketch your parallelograms and explain or show how you know their areas are equal. Then, click on the Check button to see if the two areas are indeed equal.

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Two different parallelograms $P$ and $Q$ both have an area of 20 square units. Neither parallelogram is a rectangle.

On the grid, draw two parallelograms that could be $P$ and $Q$ .

Are you ready for more?

Here is a parallelogram composed of smaller parallelograms. The shaded region is composed of four identical parallelograms. All lengths are in inches.

A green parallelogram composed of smaller parallelograms with a white parallelogram in the middle.

What is the area of the unshaded parallelogram in the middle? Explain or show your reasoning.

Lesson Summary

In this lesson, we learned about 2 important parts of parallelograms, the base and the height.

We can choose any of the four sides of a parallelogram as the base. Both the side (the segment) and its length (the measurement) are called the base.
If we draw any perpendicular segment from a point on the base to the opposite side of the parallelogram, that segment will always have the same length. We call that value the height. There are infinitely many segments that can represent the height!

Any pair of base and corresponding height can help us find the area of a parallelogram, but some base-height pairs are more easily identified than others.

We often use letters to stand for numbers. If $b$ is the length of a base of a parallelogram (in units), and $h$ is the length of the corresponding height (in units), then the area of the parallelogram (in square units) is the product of these two numbers, $b \cdot h$ . Notice that we write the multiplication symbol with a small dot instead of a $\times$ symbol. This is so that we don’t get confused about whether $\times$ means multiply, or whether the letter is $x$ standing in for a number.

When a parallelogram is drawn on a grid and has horizontal sides, we can use a horizontal side as the base. When it has vertical sides, we can use a vertical side as the base. The grid can help us find (or estimate) the lengths of the base and of the corresponding height.

Two drawings of parallelograms on a grid with b and height shown. On one picture height is vertical, on the other it is horizontal.

When a parallelogram is not drawn on a grid, we can still find its area if a base and a corresponding height are known.

A parallelogram with side lengths 10 units and 8 units. An 8-unit perpendicular segment connects one vertex of the 8 unit side to a point on the other 8 unit side.

In this parallelogram, the corresponding height for the side that is 10 units long is not given, but the height for the side that is 8 units long is given. This base-height pair can help us find that the area is 64 square units since $8 \cdot 8 = 64$ .

Regardless of their shape, parallelograms that have the same base and the same height will have the same area; the product of the base and height will be equal. Here are some parallelograms with the same pair of base-height measurements.

Four different parallelograms. Each parallelogram has a base labeled 3 and a height labeled 4.