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.

Square A, shaded. Square B identical to A, with a small shaded square removed in the middle and a small shaded square appended to its side.

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.

Four figures, each on a white square grid. 


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.

3 figures labeled A, B, C.

 

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.
Two images of t-shaped objects on a grids.
  • 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.
3 figures on grids with arrows pointing to the right between figures 1 and 2 and figures 2 and 3. 
  • 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.
Two shaded squares in a grid. Each are 6 units square and each as a 1 unit by two unit portion that is unshaded.

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.

rectangle, base = 8 centimeters, height = 4 centimeters.

Video Summary

Video VLS G6U1V1 Reasoning to Find Area (Lessons 1–3, 11) available at https://player.vimeo.com/video/443554693.

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.