# 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.

2. 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.

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

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

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

Find a way to use all of your pieces to compose a single large square. What is the area of this large square?

### 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 . . .

### 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 Entries

• area

Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.

For example, the area of region A is 8 square units. The area of the shaded region of B is $$\frac12$$ square unit.

• compose

Compose means “put together.” We use the word compose to describe putting more than one figure together to make a new shape.

• decompose

Decompose means “take apart.” We use the word decompose to describe taking a figure apart to make more than one new shape.

• region

A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere.