Lesson 2: Side Lengths and Areas

Let’s investigate some more squares.

2.1: Notice and Wonder: Intersecting Circles

What do you notice? What do you wonder?

2.2: One Square

  1. Use the circle to estimate the area of the square shown here:

    A coordinate plane with the origin labeled “O.” The x-axis has the numbers negative 6 through 6 indicated with tick marks. The y-axis has the numbers negative 6 through 7 indicated with tick marks. A square and a circle are drawn on the grid so that the circle’s circumference passes through 2 of the squares vertices. The circle’s center is the origin and it’s circumference is indicated by a dashed line that passes through the following approximate points on the axes: Negative 5 point 3 comma 0, 0 comma 5 point 3, 5 point 3 comma 0, and 0 comma negative 5 point 3. The square is tilted so that all its sides are diagonal to the coordinate grid. It has vertices at: 0 comma 0, negative 2 comma 5, 3 comma 7, and 5 comma 2. The circumference of the circle passes through the square’s vertices at negative 2 comma 5 and 5 comma 2 so that the sides of the square, extending from the origin to those 2 vertices, are within the circle. @Kia Johnson I didn't want to say that the sides of the square were the radius felt like taking away some of the cognitive demand) but felt a little wordy. REPLY 11:44 (Fixed some language, now that I am writing for same image on grid): A coordinate plane with the origin labeled “O.” The x-axis has the numbers negative 6 through 6 indicated with tick marks. The y-axis has the numbers negative 6 through 7 indicated with tick marks. A square and a circle are drawn on the plane so that the circle’s circumference passes through 2 of the squares vertices. The circle’s center is the origin and it’s circumference is indicated by a dashed line that passes through the following approximate points on the axes: Negative 5 point 3 comma 0, 0 comma 5 point 3, 5 point 3 comma 0, and 0 comma negative 5 point 3. The square is tilted so that all its sides are diagonal to the coordinate grid. It has vertices at: 0 comma 0, negative 2 comma 5, 3 comma 7, and 5 comma 2. The circumference of the circle passes through the square’s vertices at negative 2 comma 5 and 5 comma 2 so that the sides of the square, extending from the origin to those 2 vertices, are within the circle.

  2. Use the grid to check your answer to the first problem.

    A coordinate grid with the origin labeled “O.” The x-axis has the numbers negative 6 through 6 indicated with gridlines. The y-axis has the numbers negative 6 through 7 indicated with gridlines. A square and a circle are drawn on the grid so that the circle’s circumference passes through 2 of the squares vertices. The circle’s center is the origin and it’s circumference is indicated by a dashed line that passes through the following approximate points on the axes: Negative 5 point 3 comma 0, 0 comma 5 point 3, 5 point 3 comma 0, and 0 comma negative 5 point 3. The square is tilted so that all its sides are diagonal to the coordinate grid. It has vertices at: 0 comma 0, negative 2 comma 5, 3 comma 7, and 5 comma 2. The circumference of the circle passes through the square’s vertices at negative 2 comma 5 and 5 comma 2 so that the sides of the square, extending from the origin to those 2 vertices, are within the circle.

2.3: The Sides and Areas of Tilted Squares

  1. Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.

  2. Complete the tables with the missing side lengths and areas.
      side length, $s$ 0.5   1.5   2.5   3.5  
    row 1 area, $a$    1     4     9    16
      side length, $s$ 4.5   5.5   6.5   7.5  
    row 1 area, $a$   25   36   49   64
     

  3. Plot the points, $(s, a)$, on the coordinate plane shown here.

    GeoGebra Applet XF2dwtWK

  4. Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?

  5. Use the graph to approximate $\sqrt{45}$.

Summary

We saw earlier that the area of square ABCD is 73 units2.

What is the side length? The area is between $8^2 = 64$ and $9^2 = 81$, so the side length must be between 8 units and 9 units. We can also use tracing paper to trace a side length and compare it to the grid, which also shows the side length is between 8 units and 9 units. But we want to be able to talk about its exact length. In order to write “the side length of a square whose area is 73 square units,” we use the square root symbol. “The square root of 73” is written $\sqrt{73}$, and it means “the length of a side of a square whose area is 73 square units.”

We say the side length of a square with area 73 units2 is $\sqrt{73}$ units. This means that

$$\left( \sqrt{73}\right)^2 = 73$$

All of these statements are also true:

$\sqrt{9}=3$ because $3^2=9$

$\sqrt{16}=4$ because $4^2=16$

$\sqrt{10}$ units is the side length of a square whose area is 10 units2, and $\left(\sqrt{10}\right)^2=10$

There are 3 squares on a square grid, arranged in order of area, from smallest, on the left, to largest, on the right. The left most square is aligned to the grid and has side lengths of 3 with an area of 9. The middle square is tilted on the grid so that its sides are diagonal to the grid. The square is labeled with a side length of square root of 10 and an area of 10. The right most square is aligned to the grid and has side lengths of 4 with an area of 16.

Practice Problems ▶

Glossary

square root

square root

The square root of a positive number $x$, written $\sqrt{x}$, is the positive number whose square is $x$. For example, $\sqrt{4} = 2$ because $2^2 = 4$ and $2$ is positive. The square root of 0 is 0.