Lesson 3Reasoning to Find Area

Learning Goal

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

Learning Targets

  • I can use different reasoning strategies to find the area of shapes.

Lesson Terms

  • area
  • decompose
  • region

Warm Up: Comparing Regions

Problem 1

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.

Activity 1: On the Grid

Problem 1

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.

  1. a blue corner piece with 6 sides on a white square grid.
  2. a 6 by 6 blue square with a 3 by 3 white square inside on a white square grid
  3. a 6 by 6 blue square with a tilted white square inside on a white square grid
  4. a blue tilted square on a white square grid

Are you ready for more?

Problem 1

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

blank grid

Activity 2: Off the Grid

Problem 1

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

  1. two triangles sharing a base of 5 cm and both having heights of 3 cm
  2. square inscribed in another square forming triangles in the corners with side measures of 4 cm and 2cm
  3. small square of side measure 2 cm inside a larger square of side measure 5 cm

Lesson 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 a t-shaped object. The upper portion is 2 units tall and 6 units wide. The stem of the “t” is 4 units tall and 2 units wide. The second image is the same, except there is a line separating the upper portion and lower portion into two rectangles.

  • 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.

Two triangles next to each other making an irregular shape. One triangle is moved so the new shape is a rectangle.

  • 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.

The area of this rectangle is 32 square centimeters.

A blue rectangle with side lengths of 8 cm and 4 cm.