# Lesson 3

Reasoning to Find Area

Let’s decompose and rearrange shapes to find their areas.

### 3.1: Comparing Regions

Is the area of Figure A greater than, less than, or equal to the area of the shaded region in Figure B? Be prepared to explain your reasoning.

### 3.2: On the Grid

Each grid square is 1 square unit. Find the area, in square units, of each shaded region without counting every square. Be prepared to explain your reasoning.

Rearrange the triangles from Figure C so they fit inside Figure D. Draw and color a diagram of your work.

### 3.3: Off the Grid

Find the area of the shaded region(s) of each figure. Explain or show your reasoning.

### Summary

There are different strategies we can use to find the area of a region. We can:

• Decompose it into shapes whose areas you know how to calculate; find the area of each of those shapes, and then add the areas.
• Decompose it and rearrange the pieces into shapes whose areas you know how to calculate; find the area of each of those shapes, and then add the areas.
• Consider it as a shape with a missing piece; calculate the area of the shape and the missing piece, and then subtract the area of the piece from the area of the shape.

The area of a figure is always measured in square units. When both side lengths of a rectangle are given in centimeters, then the area is given in square centimeters. For example, the area of this rectangle is 32 square centimeters.

### 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.