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.
    Four drawings that each show squares inside a shape.
  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.

Images of squares and triangles being decomposed and composed to form different shapes. 

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  • 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.

    Figure A on the left composed of 8 identical shaded squares arranged in 3 rows. Figure B on the right consists of one square with a diagonal segment from corner to corner. Half of the square is shaded.
  • compose

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

    Image on left shows three separate parts of a shape; image on right shows those three parts put together to create an oval.
  • decompose

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

    Image on left shows three parts put together to create an oval; the image on the right shows the oval separated into the three parts.
  • 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.