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.  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.
 

1.2: Decomposing to Find Area

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


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1.3: Estimating Side Lengths from Areas

  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.

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.

GeoGebra Applet n2mcbeKd

Summary

The area of a square with side length 12 units is 122 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.

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.

Practice Problems ▶