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?

Blue and yellow circles that overlap. Triangle ACB drawn in overlap. AC is yellow, cb is green, ba is blue. 

 

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.” x-axis, scale -6 to 6 , by 1's. y-axis,-6 through 7, by 1's. Circle and square graphed.

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

    A coordinate grid with the origin labeled “O.” x-axis, scale -6 to 6 , by 1's. y-axis,-6 through 7, by 1's. Circle and square graphed.


One vertex of the equilateral triangle is in the center of the square, and one vertex of the square is in the center of the equilateral triangle. What is \(x\)?

Equilateral triangle slanted right, overlapping top left corner of square. Angle from bottom side of triangle to left side of square labeled x degrees.

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.

    3 squares labeled A, B, C. Square A, side length = square root of 29. Square B, side length = 18. Square C, side length = square root of 13.
  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.

     

  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.

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

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

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

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.

Video Summary

Video VLS G8U8V1 Side Lengths and Areas of Squares (Lessons 1–2) available at https://player.vimeo.com/video/521945003.

Glossary Entries

  • square root

    The square root of a positive number \(n\) is the positive number whose square is \(n\). It is also the the side length of a square whose area is \(n\). We write the square root of \(n\) as \(\sqrt{n}\).

    For example, the square root of 16, written as \(\sqrt{16}\), is 4 because \(4^2\) is 16.  

    \(\sqrt{16}\) is also the side length of a square that has an area of 16.