Unit 8, Lesson 2
Side Lengths and Areas
Grade 8
2.1
Warm-up
- What is the side length of Square A? What is its area?
- What is the side length of Square C? What is its area?
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What is the area of Square B? What is its side length? (Use tracing paper to check your answer to this.)
2.2
Activity
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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.
2.3
Activity
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Use the circle to estimate the area of the square shown here. Explain your reasoning.
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. 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. -
Use the grid to check your answer to the first problem.
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. 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.
Student Lesson Summary
The area of square is 73 units2.
Since the area is between and , the side length must be between 8 units and 9 units. We can 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.
When we want to talk about the exact side length, we can use the square root symbol. We say “the square root of 73,” which is written as and means “the side length of a square with area 73 square units.” It is also true that .
The square root of a positive number is the positive number whose square is . It is also the side length of a square whose area is . The square root of is written as .
The square root of 16 is written as . Its value is 4 because is 16.
is also the side length of a square that has an area of 16.