Lesson 1

The Areas of Squares and Their Side Lengths

Let’s investigate the squares and their side lengths.

1.1: Two Regions

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.

 

1.2: Decomposing to Find Area

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

  1. Tilted square of square forming right triangles in each corner with leg lengths of 1 and 9 units 
  2. Tilted square inside of a square forming right triangles in each corner with both leg lengths = 5 units 
  3. Tilted blue square inside of a square forming right triangles in each corner with leg lengths of 3 and 7.


Any triangle with a base of 13 and a height of 5 has an area of \(\frac{65}{2}\).

2 right triangles on a grid partitioned into red, blue, yellow, and green parts. For each, base = 13, height= 5, area= the fraction 65 over 2    

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?

1.3: Estimating Side Lengths from Areas

3 squares labeled A, B, C. Side length of A= 5. Side length of B= square root of 25. Side length of C=6.
  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.

    3 squares labeled D, E, F. Side length of D= square root of 20. Side length of E = square root of 29. Side length of F=square root of 34.

1.4: Making Squares

Use the applet to determine the total area of the five shapes, \(D\), \(E\), \(F\), \(G\), and \(H\). Assume each small square is equal to 1 square unit.

Summary

The area of a square with side length 12 units is \(12^2\) or 144 units2.

The side length of a square with area 900 units2 is 30 units because \(30^2 = 900\).

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 \(ABCD\)?

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

Tilted square ABCD with side lengths of square root of 73 units 

The outside square \(EFGH\) has side lengths of 11 units, so its area is 121 units2. The area of each of the four triangles is \(\frac12 \boldcdot 8 \boldcdot 3 = 12\), so the area of all four together is \(4 \boldcdot 12 = 48\) 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 \(ABCD\) is \(121 - 48 = 73\) or 73 units2.

Tilted square ABCD inside of square HEFG forming right triangles in each corner with leg lengths of 3 and 8 units