Lesson 1The Areas of Squares and Their Side Lengths

Learning Goal

Let’s investigate the squares and their side lengths.

Learning Targets

  • I can find the area of a tilted square on a grid by using methods like “decompose and rearrange” and “surround and subtract.”

  • I can find the area of a triangle.

Warm Up: Two Regions

Problem 1

Which shaded region is larger? Explain your reasoning.

Two quadrilaterals, labeled “A” and “B,” on square grids. Both quadrilaterals are not aligned to the horizontal or vertical gridlines. Quadrilateral “A” is on a grid that has 2 rows of 4 squares. The quadrilateral is drawn starting at the left most vertex. The second vertex is 2 squares to the right and 1 square up from the first vertex. The third vertex is 2 squares to the right and 1 square down from the second vertex. The fourth vertex is 2 squares to the left and 1 square down from the third vertex. The first vertex is 2 squares to the left and 1 square up from the fourth vertex. Quadrilateral “B” is on a grid that has 3 rows of 3 squares. The quadrilateral is drawn starting at the left most vertex. The second vertex is 1 square to the right and 2 squares up from the first vertex. The third vertex is 2 squares to the right and 1 square down from the second vertex. The fourth vertex is 1 square to the left and 2 squares down from the third vertex. The first vertex is 2 squares to the left and 1 square up from the fourth vertex.

Activity 1: Decomposing to Find Area

Problem 1

Find the area of each shaded square (in square units).

  1. A square on a grid with a blue square contained in it set an an angle so 4 triangles are formed between the edge of the blue square and outer square.
  2. A square not showing the grid with a blue square inside turned so that the edges touch the midpoints of the outer square.
  3. A square with a blue square inside oriented to make four triangles between the edge of the blue square and outer square. The triangles have a base of 3 and height of 7.

Are you ready for more?

Problem 1

Any triangle with a base of 13 and a height of 5 has an area of .

A right triangle on a grid. Inside are 2 triangles-1 blue, 1 yellow, and a green shape and red shape. A 2nd triangle has the same shapes but has a square uncovered in the middle.

Both shapes in the figure have been partitioned into the same four pieces. Find the area of each of the pieces, and verify the corresponding parts are the same in each picture. There appears to be one extra square unit of area in the right figure. If all of the pieces have the same area, how is this possible?

Activity 2: Estimating Side Lengths from Areas

Problem 1

3 squares on a grid-A, B, C. Squares A and C sit on the grid lined up with horizontal grid line. Square B is turned so that its vertices intersect with intersecting grid lines.

  1. What is the side length of square A? What is its area?

  2. What is the side length of square C? What is its area?

  3. What is the area of square B? What is its side length? (Use tracing paper to check your answer to this.)

  4. Find the areas of squares D, E, and F. Which of these squares must have a side length that is greater than 5 but less than 6? Explain how you know.

    Three squares on a grid labeled D, E, F. They appear to be of different sizes and are all tilted.

Activity 3: Making Squares

Problem 1

Use the applet to determine the total area of the five shapes, , , , , and . Assume each small square is equal to 1 square unit.

open applet in presentation mode
Print Version

Your teacher will give your group a sheet with three squares and 5 cut out shapes labeled D, E, F, G, and H. Use the squares to find the total area of the five shapes. Assume each small square is equal to 1 square unit.

Lesson Summary

The area of a square with side length units is or units².

The side length of a square with area units² is units because .

Sometimes we want to find the area of a square but we don’t know the side length. For example, how can we find the area of square ?

One way is to enclose it in a square whose side lengths we do know.

A blue square on a grid labeled ABCD and tilted.

The outside square has side lengths of units, so its area is units². The area of each of the four triangles is , so the area of all four together is units². To get the area of the shaded square, we can take the area of the outside square and subtract the areas of the 4 triangles. So the area of the shaded square is or units².

A square EFGH on a grid with a blue square ABCD contained in it set an an angle so 4 triangles are formed between the edge of the blue square and outer square.