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Lesson 2Side Lengths and Areas

Let’s investigate some more squares.

Learning Targets:

  • I can explain what a square root is.
  • I understand the meaning of expressions like \sqrt{25} and \sqrt{3} .
  • If I know the area of a square, I can express its side length using square root notation.

2.1 Notice and Wonder: Intersecting Circles

What do you notice? What do you wonder?

Two circles are overlapping. Where they overlap, point A of a triangle is created. Points B and C are on the opposite points of the two circles.

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.

Are you ready for more?

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 ?

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. The two meeting side lengths create an angle labeled "x."

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.

    Squares A, B, and C are graphed on a grid.
  2. Complete the tables with the missing side lengths and areas.
    side length,  s 0.5 1.5 2.5 3.5
    area, a 1  4  9  16
    side length,  s 4.5 5.5 6.5 7.5
    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} .

Lesson 2 Summary

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

Square ABCD is graphed on a grid.

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.

Glossary Terms

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. 

Lesson 2 Practice Problems

  1. A square has an area of 81 square feet. Select all the expressions that equal the side length of this square, in feet.

    1. \frac{81}{2}

    2. \sqrt{81}

    3. 9

    4. \sqrt{9}

    5. 3

  2. Write the exact value of the side length, in units, of a square whose area in square units is:

    1. 36
    2. 37
    3. \frac{100}{9}
    4. \frac25
    5. 0.0001
    6. 0.11

  3. Square A is smaller than Square B. Square B is smaller than Square C.

    There are 3 differently sized squares labeled, from left to right, “A,” “B” and “C.” The squares are arranged from smallest to largest, so that “A” is the smallest square and “C” is the largest.

    The three squares’ side lengths are \sqrt{26} , 4.2, and \sqrt{11} .

    What is the side length of Square A? Square B? Square C? Explain how you know.

  4. Find the area of a square if its side length is:

    1. \frac15 cm
    2. \frac37 units
    3. \frac{11}{8} inches
    4. 0.1 meters
    5. 3.5 cm

  5. Here is a table showing the areas of the seven largest countries.

    1. How much more area is there in Russia than in Canada?
    2. The Asian countries on this list are Russia, China, and India. The American countries are Canada, the United States, and Brazil. Which has the greater total area: the three Asian countries, or the three American countries?
    country area (in km2)
    Russia 1.71 \times 10^7
    Canada 9.98 \times 10^6
    China 9.60 \times 10^6
    United States 9.53 \times 10^6
    Brazil 8.52 \times 10^6
    Australia 6.79 \times 10^6
    India 3.29 \times 10^6

  6. Select all the expressions that are equivalent to 10^{\text-6} .

    1. \frac{1}{1000000}
    2. \frac{\text-1}{1000000}
    3. \frac{1}{10^6}
    4. 10^{8} \boldcdot 10^{\text-2}
    5. \left(\frac{1}{10}\right)^6
    6. \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}