Lesson 5Bases and Heights of Parallelograms

Learning Goal

Let’s investigate the area of parallelograms some more.

Learning Targets

I can identify base and height pairs of a parallelogram.
I can write and explain the formula for the area of a parallelogram.
I know what the terms “base” and “height” refer to in a parallelogram.

Lesson Terms

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

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?

Print Version

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: The Right Height?

Problem 1

Each parallelogram has a side that is labeled “base.”

Study the examples and non-examples of bases and heights of parallelograms. Then, answer the questions that follow.

Examples: The dashed segment in each drawing represents the corresponding height for the given base.

Four parallelograms showing a dashed segment with representing the corresponding height for the given base.

Non-examples: The dashed segment in each drawing does not represent the corresponding height for the given base.

Four parallelograms showing a dashed segment representing the corresponding height for the given base incorrectly.

Select all statements that are true about bases and heights in a parallelogram.

Only a horizontal side of a parallelogram can be a base.
Any side of a parallelogram can be a base.
A height can be drawn at any angle to the side chosen as the base.
A base and its corresponding height must be perpendicular to each other.
A height can only be drawn inside a parallelogram.
A height can be drawn outside of a parallelogram, as long as it is drawn at a 90-degree angle to the base.
A base cannot be extended to meet a height.

Problem 2

Five students labeled a base $b$ and a corresponding height $h$ for each of these parallelograms. Are all drawings correctly labeled? Explain how you know.

Are you ready for more?

In the applet, the parallelogram is made of solid line segments, and the height and supporting lines are made of dashed line segments. A base ( $b$ ) and corresponding height ( $h$ ) are labeled.

Experiment with dragging all of the movable points around the screen. Can you change the parallelogram so that …

its height is in a different location?
it has horizontal sides?
it is tall and skinny?
it is also a rectangle?
it is not a rectangle, and has $b = 5$ and $h = 3$ ?

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Can you create a parallelogram for each scenario below so that …

its height is in a different location?
it has horizontal sides?
it is tall and skinny?
it is also a rectangle?
it is not a rectangle, and has b=5 and h=3?

Activity 2: 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?

Lesson Summary

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 line segments that can represent the height!

Two parallelograms divided into segments overlayed with a rectangle.

Here are two copies of the same parallelogram. On the left, the side that is the base is 6 units long. Its corresponding height is 4 units. On the right, the side that is the base is 5 units long. Its height is 4.8 units. For both, three different segments are shown to represent the height. We could draw in many more!

No matter which side is chosen as the base, the area of the parallelogram is the product of that base and its corresponding height. We can check it:

$4 \times 6 = 24$ $and$ $4.8 \times 5 = 24$

We can see why this is true by decomposing and rearranging the parallelograms into rectangles.

Two parallelograms decomposed and rearranged into rectangles.

Notice that the side lengths of each rectangle are the base and height of the parallelogram. Even though the two rectangles have different side lengths, the products of the side lengths are equal, so they have the same area! And both rectangles have the same area as the parallelogram.

We often use letters to stand for numbers. If $b$ is base of a parallelogram (in units), and $h$ is 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 $x$ is standing in for a number.

In high school, you will be able to prove that a perpendicular segment from a point on one side of a parallelogram to the opposite side will always have the same length.

A parallelogram with five dashed perpendicular segments extending from the bottom of the parallelogram to the top. Each of the lines is labeled “h”.

You can see this most easily when you draw a parallelogram on graph paper. For now, we will just use this as a fact.