Finding Area by Decomposing and Rearranging

Student 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.

An 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 12\frac12 square unit.

Visual / Anchor Chart

Standards

Building On
3.MD.5.b

3.MD.C.5.b

3.MD.5.b

3.MD.C.5.b

Addressing
6.G.1

6.G.A.1

6.G.1

6.G.A.1

Building Toward
6.G.A

6.G.A

6.G.A

6.G.A