Lesson 2Finding Area by Decomposing and Rearranging

Learning Goal

Let’s create shapes and find their areas.

Learning Targets

  • I can explain how to find the area of a figure that is composed of other shapes.

  • I know how to find the area of a figure by decomposing it and rearranging the parts.

  • I know what it means for two figures to have the same area.

Lesson Terms

  • area
  • compose
  • decompose
  • region

Warm Up: What Is Area?

Problem 1

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.

    1. shape broken up into large squares
    2. shape broken up into large and small squares
    3. shape broken up into large squares and white space
    4. shape broken up into small squares

  2. Write a definition of area that includes all the information that you think is important.

Activity 1: Composing Shapes

Problem 1

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.

open applet in presentation mode

  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.

Print Version

Your teacher will give you one square and some small, medium, and large right triangles. The area of the square is 1 square unit.

  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. Trace your shape.

  3. Use your shapes to create a new shape with an area of 2 square units. Trace your shape.

  4. Use your shapes to create a different shape with an area of 2 square units. Trace your shape.

  5. Use your shapes to create a new shape with an area of 4 square units. Trace your shape.

Are you ready for more?

Problem 1

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

Activity 2: Tangram Triangles

Problem 1

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

open applet in presentation mode

  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

Print Version

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

Lesson 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.

An image of two squares, which are then decomposed into four triangles, which are then rearranged into one large triangle. Another image of a large triangle that is decomposed into four smaller triangles, and then rearranged into two squares.

  • 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 square unit.