Parallelograms

Learning Objectives

• Understand basic concepts of the meaning of area.
• Use formulas to find the area of specific types of polygons.

Introduction

Measurement is not a new topic. You have been measuring things nearly all your life. Sometimes you use standard units (pound, centimeter), sometimes nonstandard units (your pace or arm span). Space is measured according to its dimension.

• One-dimensional space: measure the length of a segment on a line.
• Two-dimensional space: measure the area that a figure takes up on a plane (flat surface).
• Three-dimensional space: measure the volume that a solid object takes up in “space.”

In this lesson, we will focus on basic ideas about area in two-dimensional space. Once these basic ideas are established we’ll look at the area formulas for some of the most familiar two-dimensional figures.

Basic Ideas of Area

Measuring area is just like measuring anything; before we can do it, we need to agree on standard units. People need to say, “These are the basic units of area.” This is a matter of history. Let’s re-create some of the thinking that went into decisions about standard units of area.

Example 1

What is the area of the rectangle below?

What should we use for a basic unit of area?

As one possibility, suppose we decided to use the space inside this circle as the unit of area.

To find the area, you need to count how many of these circles fit into the rectangle, including parts of circles.

So far you can see that the rectangle’s space is made up of whole circles. Determining the fractional parts of circles that would cover the remaining white space inside the rectangle would be no easy job! And this is just for a very simple rectangle. The challenge is even more difficult for more complex shapes.

Instead of filling space with circles, people long ago realized that it is much simpler to use a square shape for a unit of area. Squares fit together nicely and fill space with no gaps. The square below measures on each side, and it is called .

Now it’s an easy job to find the area of our rectangle.

The area is , because is the number of units of area (square feet) that will exactly fill, or cover, the rectangle.

The principle we used in Example 1 is more general.

The area of a two-dimensional figure is the number of square units that will fill, or cover, the figure.

Two Area Postulates

Congruent Areas

If two figures are congruent, they have the same area.

This is obvious because congruent figures have the same amount of space inside them. However, two figures with the same area are not necessarily congruent.

Area of Whole is Sum of Parts

If a figure is composed of two or more parts that do not overlap each other, then the area of the figure is the sum of the areas of the parts.

This is the familiar idea that a whole is the sum of its parts. In practical problems you may find it helpful to break a figure down into parts.

Example 2

Find the area of the figure below.

Luckily, you don’t have to learn a special formula for an irregular pentagon, which this figure is. Instead, you can break the figure down into a trapezoid and a triangle, and use the area formulas for those figures.

Basic Area Formulas

Look back at Example 1 and the way it was filled with unit area squares.

Notice that the dimensions are:

base (or length)

height (or width)

But notice, too, that the base is the number of feet in one row of unit squares, and the height is the number of rows. A counting principle tells us that the total number of square feet is the number in one row multiplied by the number of rows.

Area of a Rectangle

If a rectangle has base units and height units, then the area, , is square units.

Example 3

What is the area of the figure shown below?

Break the figure down into two rectangles.

Now we can build on the rectangle formula to find areas of other shapes.

Parallelogram

Example 4

How could we find the area of this parallelogram?

Make it into a rectangle

The rectangle is made of the same parts as the parallelogram, so their areas are the same. The area of the rectangle is , so the area of the parallelogram is also .

Warning: Notice that the height of the parallelogram is the perpendicular distance between two parallel sides of the parallelogram, not a side of the parallelogram (unless the parallelogram is also a rectangle, of course).

Area of a Parallelogram

If a parallelogram has base units and height units, then the area, , is square units.

Lesson Summary

Once we understood the meaning of measures of space in two dimensions—in other words, area—we saw the advantage of using square units. With square units established, the formula for the area of a rectangle is simply a matter of common sense. From that point forward, the formula for the area of each new figure builds on the previous figure. For a parallelogram, convert it to a rectangle. For a triangle, double it to make a parallelogram.

Points to Consider

As we study other figures, we will frequently return to the basics of this lesson—the benefit of square units, and the fundamental formula for the area of a rectangle.

It might be interesting to note that the word geometry is derived from ancient Greek roots that mean Earth (geo-) measure (-metry). In ancient times geometry was very similar to today’s surveying of land. You can see that land surveying became easily possible once knowledge of how to find the area of plane figures was developed.

The questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

Complete the chart. Base and height are given in units; area is in square units.

1. 	Base	Height	'Area
   1a.			?
   1b.		?
   1c.			?
   1d.		?
   1e.			?
   1f.		?

2. The carpet for a by room cost . The same kind of carpet cost for a room with a square floor. What are the dimensions of the room?

Review Answers

1. 1a.

   1b.

   1c.

   1d.

   1e.

   1f.

2. by