# Lesson 1: The Areas of Squares and Their Side Lengths

Let’s investigate the squares and their side lengths.

## 1.1: Two Regions ## 1.2: Decomposing to Find Area

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

1. 2. 3. ## 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 $12^2$ 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. 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. 