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.
- 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.
- 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.
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.
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.
Use your shapes to create a new shape with an area of 2 square units. Draw your shape and label it with its area.
Use your shapes to create a different shape with an area of 2 square units. Draw your shape and label it with its area.
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.
- The area of the small triangle is ____________ square units. I know this because . . .
- The area of the medium triangle is ____________ square units. I know this because . . .
- The area of the large triangle is ____________ square units. I know this because . . .
Here are two important principles for finding area:
If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
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.
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 means “put together.” We use the word compose to describe putting more than one figure together to make a new shape.
Decompose means “take apart.” We use the word decompose to describe taking a figure apart to make more than one new shape.
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.