# Lesson 2: Finding Area by Decomposing and Rearranging

Let’s create shapes and find their areas.

## 2.1: What is Area?

You may recall that the term area tells us something about the number of squares inside a two-dimensional shape.

1. Here are four drawings that each show squares inside a shape. Select all drawings whose squares could be used to find the area of the shape. Be prepared to explain your reasoning. 2. Write a definition of area that includes all the information that you think is important.

## 2.2: Composing Shapes

This applet has one square and some small, medium, and large right triangles. The area of the square is 1 square unit.

Click on a shape and drag to move it. Grab the point at the vertex and drag to turn it.

1. Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.

GeoGebra Applet PHTsMPDY

1. Use your shapes to create a new shape with an area of 1 square unit that is not a square. Draw your shape on paper and label it with its area.

2. Use your shapes to create a new shape with an area of 2 square units. Draw your shape and label it with its area.

3. Use your shapes to create a different shape with an area of 2 square units. Draw your shape and label it with its area.

4. Use your shapes to create a new shape with an area of 4 square units. Draw your shape and label it with its area.

## 2.3: Tangram Triangles

Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.

1. The area of the small triangle is ____________ square units. I know this because . . .
2. The area of the medium triangle is ____________ square units. I know this because . . .
3. The area of the large triangle is ____________ square units. I know this because . . .

GeoGebra Applet J4f6V5NE

## Summary

Here are two important principles for finding area:

1. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.

2. We can decompose a figure (break a figure into pieces) and rearrange the pieces (move the pieces around) to find its area.

Here are illustrations of the two principles. • Each square on the left can be decomposed into 2 triangles. These triangles can be rearranged into a large triangle. So the large triangle has the same area as the 2 squares.
• Similarly, the large triangle on the right can be decomposed into 4 equal triangles. The triangles can be rearranged to form 2 squares. If each square has an area of 1 square unit, then the area of the large triangle is 2 square units. We also can say that each small triangle has an area of $\frac12$ square unit.

## Glossary

area

#### area

The area of a two-dimensional region, measured in square units, is the number of unit squares that cover the region without gaps or overlaps.

The side length of each square is 1 centimeter. The area of the shaded region A is 8 square centimeters. The area of shaded region B is $\frac12$ square centimeters. rearrange

#### rearrange

When we decompose a figure into pieces and put them back together in a different way, we are rearranging the pieces.

compose/decompose

#### compose/decompose

Compose means “put together” and decompose means “take apart.” We use the word “compose” to describe putting several geometric figures together to make a new figure.