Circles - Austin Independent School Districtcurriculum.austinisd.org/.../M_7_Carnegie_Circles.pdf · to the length of its diameter. X 9.3 Area of a Circle 7.8.B 7.8.C 1 ... Step 2:center

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Circles

9.1 IntroductiontoCirclesCircle, Radius, and Diameter .......................................477

9.2 ButMostofAll,ILikePi!Circumference of a Circle ........................................... 483

9.3 OneMillionSides!Area of a Circle ...........................................................493

9.4 It’sAboutCircles!Unknown Measurements ............................................ 505

Crop circles

are really cool, and they come in all kinds of

complex designs. The designs usually can only be seen from the air. As nice

as they are though, they are still harmful

to crops.

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475A • Chapter 9 Circles

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Chapter 9 OverviewThis chapter develops the formula for the area of a circle. Problems are solved involving circles and circle formulas.

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9.1 Circle, Radius, and Diameter 4.6.A 1

Below Grade LevelThis lesson investigates the definition of a circle and focuses on the relationship between the radius length and diameter length of a circle. Questions ask students to use circles to construct equilateral polygons.

X

9.2 Circumference of a Circle

7.5.B7.8.C7.9.B

1

This lesson explores the relationship between the radius and diameter lengths of a circle to the circle’s circumference using string and a ruler. Pi is defined as the ratio of the circumference of a circle to the length of its diameter.

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9.3 Area of a Circle 7.8.B7.8.C 1

This lesson derives the formula for the area of a circle, and then explores the relationship between a circle’s circumference and its area using circles inscribed in various regular polygons. Questions focus students to understand that as the number of sides of the regular polygon increases, the perimeter of the polygon approaches the circumference of the inscribed circle and the area of the polygon approaches the area of the inscribed circle.

X X

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Chapter 9 Circles • 475B

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9.4 Unknown Measurements 7.9.B7.9.C 1

This lesson presents the area and circumference formulas to solve for unknown measurements in composite figures. This lesson also presents students with different problem situations, and students are to determine when it is appropriate to use the circumference formula and when it is appropriate to use the area formula.

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476 • Chapter 9 Circles

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Skills Practice Correlation for Chapter 9

LessonsProblem

SetObjective(s)

9.1 Circle, Radius, and Diameter

Vocabulary

1 – 10 Name and identify parts of circles

11 – 18 Determine if circles are congruent

19 – 20 Construct figures using congruent circles

9.2 Circumference of a Circle

Vocabulary

1 – 6 Measure the radius and circumference of circles and calculate ratios of circumference to diameter

7 – 12 Calculate values using the circumference formula

13 – 18 Calculate values using the circumference formula

9.3 Area of a Circle

Vocabulary

1 – 6 Calculate areas of circles

7 – 12 Calculate areas of annuli

13 – 16 Answer questions about a circle inscribed in an equilateral triangle

17 – 20 Answer questions about a circle inscribed in a square

21 – 24 Answer questions about a circle inscribed in a pentagon

25 – 28 Answer questions about a circle inscribed in a hexagon

9.4 Unknown Measurements1 – 6 Use the area and circumference formulas to answer

questions

7 – 12 Find the area of shaded regions

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9.1 Circle, Radius, and Diameter • 477A

Key Terms circle

center of a circle

radius of a circle

diameter of a circle

Learning GoalsIn this lesson, you will:

Define circle.

Identify the center, radius,

and diameter of a circle.

Essential Ideas

• A circle is a collection of points on the same plane equidistant from the same point. The center point is the point from which the collection of points is equidistant.

• The radius of a circle is a line segment formed by connecting a point on the circle to the center point of the circle.

• The diameter of a circle is a line segment formed by connecting two points on the circle such that the line segment passes through the center point of the circle.

• The length of a radius of a circle is half the length of the diameter of the circle.

Introduction to CirclesCircle, Radius, and Diameter

Texas Essential Knowledge and Skills for MathematicsGrade4

(6) Geometry and measurement. The student applies mathematical process standards to analyze geometric attributes in order to develop generalizations about their properties. The student is expected to:

(A) identify points, lines, line segments, rays, angles, and perpendicular and parallel lines;

MaterialsStraightedgeRulerCompass

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477B • Chapter 9 Circles

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OverviewThe terms circle, radius of a circle, diameter of a circle, and center of a circle are reviewed. Students

explain the relationship between the length of the radius of a circle and the length of the diameter of a

circle. Students conclude congruent circles have congruent radii and congruent diameters. They will

use circles to construct equilateral polygons.

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Warm Up

1. Draw a circle with the length of a radius equal to 4 centimeters.

4 in.

2. Draw a circle with the length of a diameter equal to 4 centimeters.

4 in.

3. How is the circle in Question 1 similar to the circle in Question 2?

The circle drawn in Question 1 and the circle drawn in Question 2 are both collections of

points that were drawn equidistant from a center point.

4. How is the circle in Question 1 different than the circle in Question 2?

The circle drawn in Question 1 is larger or has a greater area than the circle drawn in

Question 2. The length of the radius of the circle in Question 1 is 4 centimeters, and the

length of the radius of the circle in Question 2 is 2 centimeters.

9.1 Circle, Radius, and Diameter • 477C

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477D • Chapter 9 Circles

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9.1 Circle, Radius, and Diameter • 477

You stop a friend in the hallway to ask about his or her weekend. As you are

talking, another friend shows up and joins the conversation. Soon, another and

then another and then another person joins in. Before long, your group will form a

shape without even thinking about it. Your group will probably form a circle.

Try and notice this the next time it happens—either to you or in another group.

Why do you think people naturally form a circle when talking?

Key Terms circle

center of a circle

radius of a circle

diameter of a circle


Define circle.

Identify the center, radius,

and diameter of a circle.

IntroductiontoCirclesCircle, Radius, and Diameter

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Problem 1 Definition of a Circle

Everyone can identify a circle when they see it, but defining a circle is a bit harder. Can

you define a circle without using the word round? Investigating how a circle is formed will

help you mathematically define a circle.

Step 1: In the middle of the space below, draw a point and label the point B.

A C

B

Step 2: Use a centimeter ruler to locate and draw a second point that is exactly 5 cm

from point B. Label this point A.

Step 3: Locate a third point that is exactly 5 cm from point B. Label this point C.

Step 4: Repeat this process until you have drawn at least ten distinct points that are

each exactly 5 cm from point B.

1. How would you describe this collection of points in relation to point B?

This collection of points is equidistant from point B.

2. How many other points could be located exactly 5 cm from point B?

An infinite number of points could be located exactly 5 cm from point B.

3. All of the points you have drawn are on the same plane. Are there other points that

are 5 cm from point B that are not on this plane? If so, describe their location.

There are an infinite number of points not on this plane located exactly 5 cm from

point B. They are above and below the plane.

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Share Phase, Questions 1 through 3

• Are all of the points you have drawn considered on the circle or in the circle?

• What is the difference between a point located on a circle and a point located in a circle?

Problem 1Students draw a locus of coplanar points equidistant from a given point and write a definition for a circle.

MaterialsRuler

Grouping

• Ask a student to read the introduction to Problem 1 aloud. Discuss the information as a class.

• Have students complete Questions 1 through 6 with a partner. Then share the responses as a class.

In this chapter students will reference the different parts of a circle. Create a graphic organizer to create a visual connection for the words circle, center of circle, diameter, radius, and annulus. Giving students printed definitions of each word to add to the organizer as the word is encountered in the text provides additional support.

In this chapter students

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4. The shape formed by connecting all points located 5 cm from point B on the same

plane is two-dimensional. What is the name of this shape?

A circle is formed by connecting all points located 5 cm from point B on the

same plane.

5. The solid formed by connecting all points located 5 cm from point B is

three-dimensional. What is the name of this solid?

A sphere is formed by connecting all points located 5 cm from point B.

6. Define the term circle without using the word round.

A circle is a collection of points on the same plane equidistant from the

same point.

Problem 2 Parts of a Circle

Use the circle shown to answer each question.

F

B

E

C

H

GD

A

A circle is a collection of points on the same plane equidistant from the same point. The

center of a circle is the point from which all points on the circle are equidistant. Circles

are named by their center point.

1. Name the circle.

The circle shown is Circle B. Point B is the center of the circle.

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• Are circles considered polygons? Why or why not?

• What are some familiar objects in your house that are shaped like a circle?

• What are some familiar objects in your house that are shaped like a sphere?

• What shape is an inflated basketball?

• What shape is a deflated basketball?

Problem 2 Students review naming a circle, and identifying the radius and the diameter of a circle. They will explain the relationship between the length of the radius of a circle and the length of the diameter of a circle.

GroupingHave students complete Questions 1 through 5 with a partner. Then share the responses as a class.

Share Phase, Question 1Can a circle have more than one center point? Explain.


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The radius of a circle is a line segment formed by connecting a point on the circle and

the center of the circle. The diameter of a circle is a line segment formed by connecting

two points on the circle such that the line segments passes through the center point.

2. Identify a radius of the circle.

Answers will vary.

Line segment AB is a radius of Circle B.

3. Identify a diameter of the circle.

Answers will vary.

Line segment FG is a diameter of Circle B.

4. Are all radii of this circle the same length? Explain your reasoning.

All radii of the same circle must be the same length because all of the points on

the circle are equidistant from the center point, and a radius is formed by

connecting a point on the circle with the center point of the circle.

5. What is the relationship between the length of a radius and the length of a diameter?

The length of a radius is half the length of a diameter because two radii form each

diameter. It can also be said that the length of a diameter is twice the length of

a radius.

Problem 3 Using Congruent Circles

Recall that congruent means “the same size and the same shape.”

1. If Circle A is congruent to Circle B, what can you conclude about the lengths of the

radii in Circle A and Circle B?

The radii of the congruent circles are the same length.

2. If Circle A is congruent to Circle B, what can you conclude about the lengths of the

diameters in Circle A and Circle B?

The diameters of the congruent circles are the same length.

The plural of radius is

radii.

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• How many radii are in a circle?

• How many diameters are in a circle?

• How are radii different from diameters?

• How are radii similar to diameters?

Problem 3Analyzing congruent circles, students conclude the radii of the circles are congruent, and the diameters of the circles are congruent.



• If two circles are congruent, do they share the same center point? Explain.

• How do you determine if two circles are congruent?

• What is one way to determine two circles are not congruent?

• How many degrees are in a circle?

• If all circles are equal to 360°, why aren’t all circles congruent?

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3. Dione knows Circle D is congruent to Circle E. What is Dione’s reasoning?

D E

Circle D is congruent to Circle E because radius DE is a radius of both circles and

if two circles have the same radius, the circles are congruent.

Problem 4 Circles Related to Equilateral Polygons

Mr. Graham was explaining to his students how circles are directly related to polygons.

The class was discussing how circles do not have sides and polygons have sides, so it

was not clear to them how circles and polygons had much of anything in common. Mr.

Graham told his students to use only their compasses and try to discover a relationship.

1. Jeff discovered he could construct an equilateral triangle using two congruent circles.

Jeff’s drawing is partially shown.

A B

C

Complete Jeff’s work to show the equilateral triangle. Explain your reasoning.

If Circle A is congruent to Circle B, then the radii in both circles are the same

length. Line segments AB, BC, and CA form Triangle ABC and are all radii of two

congruent circles.

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Problem 4Students construct an equilateral triangle, a rhombus, and a regular hexagon using a compass.

MaterialsCompass

Straightedge


Share Phase, Question 1

• Is there more than one way Jeff can connect the points to form an equilateral triangle? Explain.

• Is line segment AB a radius of circle A?

• Is line segment AB a radius of circle B?

• What can you conclude about two circles that share the same radius?

• Can you draw a second radius of circle A by connecting any existing points?

• Can you draw a second radius of circle B by connecting any existing points?


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2. Jackie said that she could construct an equilateral quadrilateral using three congruent

circles. Jackie’s work is partially shown.

DC

BA

Complete Jackie’s work to show the equilateral quadrilateral. Explain your reasoning.

If Circle A, Circle B, and Circle C are all congruent, then the radii in the three circles

are the same length. Line segments AB, BC, CD, and DA form the quadrilateral

ABCD and are all radii of three congruent circles.

3. Use your compass to construct a hexagon with all congruent sides.

Explain how you did it.

A B

CGD

FE

Circle A, Circle B, Circle C, Circle E, and Circle F are all drawn congruent, and each

circle passes through another circle’s center point. Line segments AB, BG, GF, FE,

ED, and DA are all radii of the circles. Therefore they are all congruent. The

hexagon formed by these sides is a regular hexagon.

Two circles make the triangle.

Three circles make the quadrilateral. How many

circles will I need here?

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Share Phase, Questions 2 and 3

• What is another name for this equilateral quadrilateral?

• Is another name for this equilateral quadrilateral a square? Why or why not?

• Did you need to draw additional circles to form the equilateral quadrilateral? Explain.

• What do you know about an equilateral hexagon?

• Did you need to draw additional circles to form the equilateral hexagon? Explain.

• How many additional circles did you need to draw to form an equilateral hexagon?

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Follow Up

AssignmentUse the Assignment for Lesson 9.1 in the Student Assignments book. See the Teacher’s Resources

and Assessments book for answers.

Skills PracticeRefer to the Skills Practice worksheet for Lesson 9.1 in the Student Assignments book for additional

resources. See the Teacher’s Resources and Assessments book for answers.

AssessmentSee the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 9.

Check for Students’ Understanding 1. Using only a compass and a straightedge, construct a right triangle.

A

BC

2. How many circles were needed for the construction?

I used two circles to construct a right triangle ABC.

3. Describe the construction.

I began with drawing a line segment. I used each endpoint of the line segment as the center

point of a circle and drew two circles labeling one endpoint, point B. I drew the line segment

formed by the two points at which the circles intersected each other. I labeled the upper point

of the intersecting circles point A, and I labeled the point at which the two line segments

intersected each other point C. I connected points A, B, and C to form right triangle ABC.

9.1 Circle, Radius, and Diameter • 482A

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9.2 Circumference of a Circle • 483A

Key Term pi


Measure the circumference of a circle.

Explore the relationship between the diameter and the

circumference of a circle.

Write a formula for the circumference of a circle.

Use a formula to determine the circumference of a circle.

Essential Ideas

• The circumference of a circle is the distance around the circle.

• The ratio of the circumference of a circle to the diameter of a circle is approximately 3.14 or pi.

• The formula for calculating the circumference of a circle is C 5 dπ or C 5 2πr where C is the circumference of a circle, d is the length of the diameter of the circle, r is the length of the radius of the circle, and π is represented using the approximation 3.14.


(5) Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to:

(B) describe π as the ratio of the circumference of a circle to its diameter

But Most of All, I Like Pi!Circumference of a Circle

(8) Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume. The student is expected to:

(C) use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas

(9) Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems. The student is expected to:

(B) determine the circumference and area of circles

MaterialsStringRulerStraightedgeCompass

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OverviewStudents use a ruler and string to explore the circumference of a circle and its relationship to the

length of the radius of a circle and the length of the diameter of a circle. Next, they will draw their

own circles and perform the measurements to determine the ratio, again, to show it is consistent with

the previous findings. In the last activity, students use the formulas C 5 dπ and C 5 2πr to solve for

unknown measurements.

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Warm Up

1. Use a string and a centimeter ruler to determine the circumference of this circle.

a. How does your measure of the circumference compare to your classmates’ answers?

The answers are not exactly the same, but they are close.

b. Can a circle have more than one circumference?

A circle has only one circumference.

c. Should everyone’s measurements be the same? Why or why not?

Using a piece of string to measure creates small degrees of measurement error. You cannot

get an exact measurement using a piece of string.

9.2 Circumference of a Circle • 483C

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9.2 Circumference of a Circle • 483

ButMostofAll,ILikePi!Circumference of a Circle

Key Term pi


Measure the circumference of a circle.

Explore the relationship between the diameter and the


Write a formula for the circumference of a circle.

Use a formula to determine the circumference of a circle.

Beginning in about the 1970s, people in many different countries began

reporting formations formed in fields, created by flattening down crops in certain

ways. These came to be known as crop circles.

At first, people thought that weather or even aliens were creating these

formations, but it turned out that groups of people would go into fields at night

and create the crop circles themselves. Many of these formations are extremely

complex and beautiful.

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Problem 1 Measuring the Circumference of a Circle

Recall that the circumference of a circle is the distance around the circle.

Let’s explore circles.

Use a string and a centimeter ruler to measure the radius and circumference of each circle.

1.

A

B 2.

A

B

3.

C

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Problem 1Students use a ruler and a string to measure the circumference of five different circles. They measure the radius and diameter of each circle and organize all data in the table provided and then determine the ratio of the circumference of a circle : the diameter of a circle. Students will average their answers to get closer to a 3.14 ratio.

MaterialsStringRuler



• Explain how you used your string and ruler to determine the circumference.

• Were any of the circles harder to measure than the others? Why or why not?

• Can you think of a better way to determine circumference than using string and a ruler?

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4.

D

5.

E

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6. Record your measurements from Questions 1 through 5 in the table and complete

the table.

Circle Radius Diameter Circumference Circumference ______________ Diameter

Circle A

Circle B

Circle C

Circle D

Circle E

7. Average the answers in the last column.

Answers will vary. (¯ 3.14)

8. How does your answer to Question 7 compare to your classmates’ answers?

The answers are close.

9. What would explain why everyone did not get the

same answer?

The measurements of the circumference are not exact.

10. Average all of your answers to Question 7.

The average should be close to 3.14.

Do you see any

patterns?

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• Do you see any patterns in your table?

• Is there a way to get a more exact answer when using the string and ruler?

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Problem 2 Construct Your Own Circles

Use a compass to construct five of your own circles and measure the radius and

circumference of each circle.

1.

A

2.

B

3.

C

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Problem 2Students use a compass to construct five circles of their choice. Similar to the last problem, they use string and a ruler to determine the circumference, measure the length of the radius and the length of the diameter of the circle, and determine the ratio of the circumference of a circle to the diameter of a circle.

MaterialsCompassStringRuler



• Can a circle have more than one center point? Explain.

• How many radii are in a circle?

• How many diameters are in a circle?

• How are radii different than diameters?

• How are radii similar to diameters?

• How did you decide the size of each of your circles?

• Do you think drawing your own circles will have an effect on the measure of the radii and diameters?


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4.

D

5.

E

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6. Record your measurements in the table and complete the table.

Your students’ answers will vary.


Circle A

Circle B

Circle C

Circle D

Circle E

7. Average the answers in the last column.

Answers will vary. (¯ 3.14)

8. How does your answer to Question 7 compare to your

classmates’ answers?

The answers are close.

9. Average all of your answers to Question 7.

The average should be close to 3.14.

10. What symbol is used to represent the ratio of the circumference of a circle to the

diameter of the circle?

π is the symbol used to represent the ratio of the circumference of a circle to the

diameter of the circle.

How does this table

compare to the last one?

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• Do you see any patterns in your table?

• How does this table compare to the table in the previous problem?

• Did you find your answers to be more or less exact in this problem than in the last problem? Why do you think that was?

• Do you think you have enough information to prove that all circles have a ratio of the circumference to the diameter of 3.14? Why or why not?


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• If you know the radius, how can you determine the circumference?

• If you know the diameter, how can you determine the circumference?

• If you know the circumference, how can you determine the diameter?


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Problem 3 Circumference Formula

The number pi (π) is the ratio of the circumference of a circle to its diameter. That is,

pi 5 circumference of a circle _______________________ diameter of a circle

or π 5 C __ d

, where C is the circumference of the circle, and

d is the diameter of the circle. The number π has an infinite number of decimal digits that

never repeat. Some approximations used for the exact value π are 3.14 and 22 ___ 7

.

1. Use this information to write a formula for the circumference of a circle, where d

represents the diameter of a circle and C represents the circumference of a circle.

C 5 πd

2. Rewrite the formula for the circumference of a circle, where r represents the radius of

a circle and C represents the circumference of a circle.

C 5 2πr

3. The diameter of a circle is 4.5 centimeters. Compute the circumference of the circle

using the circumference formula. Let π 5 3.14.

C 5 dπ

C ¯ (4.5)(3.14) ¯ 14.13 centimeters

4. The radius of a circle is 6 inches. Compute the

circumference of the circle using the circumference

formula. Let π 5 3.14.

C 5 2πr

C ¯ (2)(3.14)(6) ¯ 37.68 inches

5. The circumference of a circle is 65.94 feet. Compute the diameter

of the circle using the circumference formula. Let π 5 3.14.

C 5 dπ

65.94 ¯ (d)(3.14)

d ¯ 21 feet

Whenever you use 3.14 for pi all

your answers are approximates.

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Problem 3Pi (π) is defined as the ratio of the circumference of a circle to the length of its diameter. Students use the formulas C 5 dπ and C 5 2πr to solve for radii, diameters, and circumferences.



• Explain the relationship between the formula C 5 πd and C 5 2πr.

• If you know the diameter, how can you determine the radius?

• If you know the radius, how can you determine the diameter?

English Language Learners may need help differentiating between pi and pie as the spoken word sounds the same. Display both terms for students. Connect the word pie with the food, and the mathematical term pi to the ratio of circumference to diameter. Guide students in creating a Frayer Model for the term pi.

English Language

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6. The circumference of a circle is 109.9 millimeters. Compute the radius of the circle

using the circumference formula. Let π 5 3.14.

C 5 2πr

109.9 ¯ (2)(3.14)(r) ¯ 17.5 millimeters

7. What is the minimum amount of information needed to compute the circumference

of a circle?

To compute the circumference of a circle, I must know the length of the radius of

the circle or the length of the diameter of the circle.

Be prepared to share your solutions and methods.


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• If you know the circumference, how can you determine the radius?

• If you know the diameter, can you determine the circumference? Explain.

• If you know the radius, can you determine the circumference? Explain.

• If you know the circumference, can you determine the diameter? Explain.

• If you know the circumference, can you determine the radius? Explain.


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Follow Up






Check for Students’ Understanding 1. Use a string and a centimeter ruler to determine the circumference of the circle shown. What

might account for measurement error?

Fitting the string around the edge of the circle and then trying to straighten it out to measure

with a ruler might is enough to account for measurement error.

2. Use a centimeter ruler to measure the length of the radius of the circle. What might account for

measurement error?

The center point of the circle is large enough to account for measurement error.

3. Use the formula C 5 2πr to determine the circumference of the circle.

Answers will vary.

C 5 2πr

C 5 2(3.14)(2.6)

C < 16.328 cm

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4. Compare the circumference determined in part (b) to the circumference determined when you

used the string.

Answers will vary.

5. Use a centimeter ruler to measure the length of the diameter of the circle. What might account for

measurement error?

The actual width of the curve defining the circle is thick enough to account for measurement

error.

6. Use the formula C 5 dπ to determine the circumference of the circle.

Answers will vary.

7. Compare the circumference determined in part (e) to the circumference determined when you

used the string.

Answers will vary.

C 5 dπ

C 5 (5.3)(3.14)

C < 16.642 cm

8. Which circumference do you think is more accurate? Explain your reasoning.

Answers will vary.

I think the circumference determined by using the diameter formula is more accurate because

I did not use a string to measure the radius or the diameter and I did not measure a distance

from the center point of the circle.

9.2 Circumference of a Circle • 492A

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9.3 Area of a Circle • 493A

Key Term concentric circles

annulus

inscribed circle


Explore the circumference and area of circles inscribed

in regular polygons.

Write a formula for the area of a polygon in terms of the

perimeter.

Explore the relationship between the circumference of

a circle and the area of a circle.

Essential Ideas

• If a circle is divided into equal parts, separated, and rearranged to resemble a parallelogram, the area of a circle can be approximated by using the formula for the area of a parallelogram with a base length equal to half the circumference and a height equal to the radius.

• The formula for calculating the area of a triangle is

A 5 1 __ 2 bh where A is the area of a triangle, b is the

length of the base of the triangle, and h is the height of the triangle.

• An inscribed circle is a circle that fits exactly within the boundaries of another shape.

• The formula for calculating the area of a circle is

A 5 1 __ 2 Cr where A is the area of a circle, C is the

circumference of the circle, and r is the length of the radius of the circle.

• The formula for calculating the area of a circle is A 5 πr 2 where A is the area of a circle, r is the length of the radius of the circle, and π is represented using the approximation 3.14.

One Million Sides!Area of a Circle


(8) Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume. The student is expected to:

(C) use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas



MaterialsStraightedgeRuler

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OverviewStudents derive the formula for the area of a circle by dividing the circle into equal parts, separating

the parts, and rearranging the parts so that they resemble a parallelogram. Students use the formula

for the area of a parallelogram with a base length equal to half the circumference and a height equal to

the radius to write a formula for the area of a circle.

A circle is inscribed in several regular polygons and in each case, students determine that the area

of these polygons can be calculated using the formula A 5 1 __ 2

Pr where A is the area of a regular

polygon, P is the perimeter of the polygon, and r is the length of the radius of the inscribed circle. As

the number of sides of the regular polygon increases, the regular polygon approaches the shape of a

circle, so they apply the same formula to the circle replacing the P representing perimeter of a regular

polygon to a C representing circumference of a circle. It is from this formula they derive the formula

A 5 πr 2 where A is the area of a circle, r is the length of the radius of the circle, and π is represented

using the approximation 3.14.

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Warm Up

1. Determine the area of the triangle shown.

16 m

8 m

A 5 1 __ 2

bh

A 5 1 __ 2

(8)(16)

A 5 64

The area of the triangle is 64 square meters.

2. Determine the area of the regular pentagon

shown.

12 mm

10 mm

A 5 1 __ 2

bh

A 5 1 __ 2

(12)(10)

A 5 60

The area of the triangle is 300 square

millimeters.

3. How did knowing how to determine the area of a triangle help you calculate the area of the

regular pentagon?

Knowing how to determine the area of a triangle was used to calculate the area of the regular

pentagon because I divided the pentagon into five congruent triangles and only had to

calculate the area of one triangle and multiply the area by five to determine the area of the

regular pentagon.

9.3 Area of a Circle • 493C

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9.3 Area of a Circle • 493

Key Terms concentric circles

annulus

inscribed circle


Explore the circumference and area of circles inscribed

in regular polygons.

Write a formula for the area of a polygon in terms of the

perimeter.

Explore the relationship between the circumference of

a circle and the area of a circle.

ONEMillionSides!Area of a Circle

How important are circles to architecture? Well this importance can be seen in

many structures you have seen either in your towns or cities, and especially the

many Gothic-style cathedrals in Europe. The architecture relies on buttresses,

actually called flying buttresses which use circles.

To understand flying buttresses, you need to know what a buttress is. A buttress

is architecture that is built against, or projecting from, a building’s walls to offer

support. However, the flying buttress uses the concepts of circles to offer

significant support to a structure’s roof and walls. In fact, when you see the flying

buttresses in Gothic cathedrals, you can almost visualize a circle within each arch.

Do you think that circles play vital roles in other architectural concepts?

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Problem 1 Area of Circles

1. Eight diameters divide a circle into 16 equal parts, as shown in Figure 1. Figure 2

shows the 16 parts separated. If you flip every other part vertically and place all

16 parts side-by-side, it will look like Figure 3.

Figure 3

Figure 1 Figure 2

a. What type of polygon does Figure 3 most closely resemble?

Figure 3 most closely resembles a parallelogram.

b. Represent the approximate base length and height of Figure 3 in terms of the

radius and circumference of the circle.

The length of the base b is approximately equal to half of the circumference of

the circle C, or b ¯ 1 __ 2

C.

The height h is approximately equal to the radius of the circle r, or h ¯ r.

c. Use your answers to part (b) to determine the formula for the area of Figure 3.

A 5 bh

5 1 __ 2

C ? r

5 1 __ 2

(2p r)r

5 p r 2

d. How does the area of Figure 3 compare to the area of the circle?

The area for Figure 3 is about the same as the area of the circle.

e. Write a formula for the area of a circle.

The formula for the area of a circle is A 5 p r2.

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Problem 1A circle is drawn with 8 diameters dividing the circle into 16 equal sections. These sections are cut into individual pieces and rearranged to resemble a parallelogram. The students will conclude the base of the parallelogram is one-half the circumference of the circle and the height is the radius of the circle. With this information, they are able to conclude the area of the parallelogram is the same as the area of the circle.

Grouping

• Have a student read the beginning of Question 1 aloud and make sure that the students understand the context.

• Have students complete Questions 1 and 2 with a partner. Then share the responses as a class.


• What would Figure 3 look like if the circle was divided into 8 parts instead of 16?

• What would Figure 3 look like if the circle was divided into 32 parts instead of 16?

• Would the formula you wrote in part (e) be the same if you divided the circle into 8 parts? Explain.

• Would the formula you wrote in part (e) be the same if you divided the circle into 32 parts? Explain.

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• What units did you use in your answer to part (a) and why?

• Would your answer to part (b) be the same if the radius of the circle was 2 centimeters instead of 1 centimeter?

GroupingHave students complete Questions 3 and 4 with a partner. Then share the responses as a class.


• Can you think of some real world examples of concentric circles?

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2. Suppose that the radius of a circle is 1 centimeter.

a. Calculate the area of the circle.

A 5 p r 2

A ¯ 3.14(1 ) 2

A ¯ 3.14

The area of the circle is approximately 3.14 square centimeters.

b. If the radius is doubled, what effect will this have on the area?

A 5 p r 2

A ¯ 3.14(2 ) 2

A ¯ 3.14 3 4

A ¯ 12.56

The area of the circle will be quadrupled.

Concentric circles are circles that share the same center.

3. Use a compass to draw two concentric circles. Then, shade the region in between, or

bounded by, the two concentric circles.


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• What mathematical operation is required to calculate the area of an annulus?


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An annulus is the region bounded by two concentric circles.

4. For the annulus shown, R represents the radius of the larger circle and r represents the

radius of the smaller circle. Suppose that R 5 8 centimeters and r 5 3 centimeters.

Calculate the area of the annulus.

R

r

Area of the large circle: Area of the small circle:

A 5 p R 2 A 5 p r 2

A 5 p ( 8 2 ) 5 64p A 5 p ( 3 2 ) 5 9p

Area of the annulus: 64p 2 9p 5 55p ¯ 172.7c m 2

The area of the annulus is approximately 172.7 square centimeters.

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Problem 2 Three Sides

Recall that the area formula for a triangle is A 5 1 __ 2

bh, where b is the length of the base of

the triangle, and h is the height of the triangle.

An inscribed circle is a circle that fits exactly within the boundaries of another shape. It is

the largest possible circle that will fit inside a plane figure.

Inscribed Circle D intersects the equilateral triangle at the midpoint of each side. The

radius of the circle is r, and the length of each side of the triangle is s, as shown.

D

s s

s

r

1. Draw three line segments from the center point of the circle to each corner of the

triangle to form three congruent triangles. How is the radius of the circle, r, related to

the three triangles?

The radius of the circle is also the height of each triangle.

2. Write a formula to describe the area of each of the three triangles you drew.

A 5 ( 1 __ 2

s ) r

3. Write a formula to describe the area of the large triangle.

A 5 3 ( 1 __ 2

sr ) or A 5 3 __ 2

sr

4. Write a formula to describe the perimeter of the

large triangle.

P 5 3s

5. Write a formula to describe the area of the large triangle in

terms of the perimeter.

If P 5 3s, then s 5 P __ 3

. So, A 5 1 __ 2

( P __ 3

) (3r) or A 5 1 __ 2

(Pr)

You know the area of 1 small triangle, and 3 small

triangles are all in the large triangle so...


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Problem 2The definition of an inscribed circle is given. A circle inscribed in an equilateral triangle is shown. Students write formulas to calculate the area of each of the three triangles the equilateral triangle is divided into, the perimeter of the triangle, and the area of the triangle in terms of the perimeter of the triangle (A 5 1 __

2 Pr).

MaterialsStraightedge

Grouping

• Ask a student to read the introduction to Problem 2 aloud. Discuss the definition as a class.

• Have students complete Questions 1 through 5 with a partner. Then share the responses as a class.


• How do we know that all three triangles are congruent?

• What is the height of each of the three triangles?

• What is the length of the base of each of the three triangles?

• Can you draw two triangles that have the same length base, and the same height, that are not congruent?

• Why can we replace the letter b in the formula A 5 1 __ 2 bh with the letter s?


Providing a visual support and explanation for the vocabulary word inscribed will help the English Language Learners with the problems in this section. Refer students to the image of the inscribed circle. Explain that an inscribed circle is inside the triangle and touches the middle of each side. Check for student understanding by having students draw other inscribed circles.

Providing a visual support

continued on the next page

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• How do we know that all four triangles are congruent?

• What is the height of each of the four triangles?

• What is the length of the base of each of the four triangles?


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Problem 3 Four Sides

Inscribed Circle D intersects the square at the midpoint of each side. The radius of the

circle is r, and the length of each side of the quadrilateral is s, as shown.

s

s

s s

r

D

1. Draw four line segments from the center point of the circle to each corner of the

quadrilateral to form four congruent triangles. How is the radius of the circle, r, related to

the four triangles?


2. Write a formula to describe the area of each of the four triangles you drew.

A 5 1 __ 2

sr or A 5 1 __ 2

(s)(r)

3. Write a formula to describe the area of the quadrilateral.

A 5 ( 1 __ 2

sr ) 4 or A 5 2sr

4. Write a formula to describe the perimeter of the quadrilateral.

P 5 4s

5. Write a formula to describe the area of the quadrilateral

in terms of the perimeter.

If P 5 4s, then s 5 P __ 4

. So, A 5 2 ( P __ 4

) r or A 5 1 __ 2

Pr

Think about the relationship between

the area of the triangles you drew and the area of the quadrilateral.

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• Why can we replace the letter h in the formula A 5 1 __

2 bh with the letter r ?

• How many small triangles are in the large triangle?

• How do you determine the perimeter of the large triangle?

Problem 3A circle inscribed in a square is shown. Students write formulas to calculate the area of each of the four triangles the square is divided into, the area of the quadrilateral, the perimeter of the quadrilateral, and the area of the quadrilateral in terms of the perimeter (A 5 1 __

2 Pr).


continued on the next page

Use the illustration to prompt students in identifying and naming parts of the circle. Write the following questions or prompts on slips of paper. Have students randomly draw a slip of paper and use the visuals to answer the question or complete the statement.

•Where is the center of the circle?

•This is a picture of a(n) circle.

•Can you identify the radius?

•Can you show me the area of the figure?

•Can you show me the perimeter of the figure?

Use the illustration to

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Problem 4 Five Sides

Inscribed Circle D intersects the regular pentagon at the midpoint of each side. The radius

of the circle is r, and the length of each side of the pentagon is s, as shown.

r

D

ss

ss

s

1. Draw five line segments from the center point of the circle to each corner of the

pentagon to form five congruent triangles. How is the radius of the circle, r,

related to the five triangles?


2. Write a formula to describe the area of each of the five

triangles you drew.

A 5 1 __ 2

sr or A 5 1 __ 2

(s)(r)

3. Write a formula to describe the area of the pentagon.

A 5 5 ( 1 __ 2

sr ) or A 5 ( 5 __ 2

sr )

4. Write a formula to describe the perimeter of the pentagon.

P 5 5s

5. Write a formula to describe the area of the pentagon in terms of

the perimeter.

If P 5 5s, then s 5 P __ 5

. So, A 5 ( 5 __ 2

) ( P __ 5

) (r) or A 5 1 __ 2

Pr

What relationship are

you thinking about while describing the area of the

pentagon?

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• Why can we replace the letter b in the formula A 5 1 __

2 bh with the letter s?



• How many small triangles are in the square?

• How do you determine the perimeter of the square?

Problem 4A circle inscribed in a regular pentagon is shown. Students write formulas to calculate the area of each of the five triangles the regular pentagon is divided into, the area of the pentagon, the perimeter of the pentagon, and the area of the pentagon in terms of the perimeter (A 5 1 __

2 Pr).




• How do we know that all five triangles are congruent?

• What is the height of each of the five triangles?

• What is the length of the base of each of the five triangles?

• Why can we replace the letter b in the formula A 5 1 __ 2 bh with the letter s?

• Why can we replace the letter h in the formula A 5 1 __ 2 bh with the letter r ?

• How many small triangles are in the pentagon?

• How do you determine the perimeter of the pentagon?


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Problem 5 Six Sides

Inscribed Circle D intersects the regular hexagon at the midpoint of each side. The radius

of the circle is r, and the length of each side of the hexagon is s, as shown.

r

s

s

D

s s

ss

1. Draw six line segments from the center point of the circle to each corner of the

hexagon to form six congruent triangles. How is the radius of the circle, r,

related to the six triangles?

The radius of the circle is also the height of

each triangle.

2. Write a formula to describe the area of each of the six

triangles you drew.

A 5 1 __ 2

sr

3. Write a formula to describe the area of the hexagon.

A 5 6 ( 1 __ 2

sr ) A 5 3sr

How did you use a similar strategy for each of these

problems?

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Problem 5A circle inscribed in a regular hexagon is shown. Students write formulas to calculate the area of each of the six triangles the regular hexagon is divided into, the area of the hexagon, the perimeter of the hexagon, and the area of the hexagon in terms of the perimeter (A 5 1 __

2 Pr ).




• How do we know that all six triangles are congruent?

• What is the height of each of the six triangles?

• What is the length of the base of each of the six triangles?

• Why can we replace the letter b in the formula A 5 1 __

2 bh with the letter s?



• How many small triangles are in the hexagon?

• How do you determine the perimeter of the hexagon?

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4. Write a formula to describe the perimeter of the hexagon.

P 5 6s

5. Write a formula to describe the area of the hexagon in terms of the perimeter.

If P 5 6s, then s 5 P __ 6

. So, A 5 3 ( P __ 6

) (r) or A 5 1 __ 2

Pr

Problem 6 A Million Sides

Use your answers to Question 5 in each of the last four

problems to generalize.

1. If a regular polygon has one million sides, what

formula could be used to compute the area of the

polygon in terms of the perimeter?

A 5 1 __ 2

Pr, where r is the radius of the inscribed circle.

2. If a regular polygon has n sides, what formula could be

used to compute the area of the polygon in terms of

the perimeter?

A 5 1 __ 2

Pr, where r is the radius of the inscribed circle.

I need to go back and look for a pattern.

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Problem 6Students generalize the results of previous problems by writing a formula to determine the area of a polygon with a million sides in terms of the perimeter of the polygon. Next, they will write a formula to determine the area of a regular polygon with n sides, in terms of the perimeter.



• What formula did you use to calculate the area of a square in terms of the perimeter?

• What formula did you use to calculate the area of a regular pentagon in terms of the perimeter?

• What formula did you use to calculate the area of a regular hexagon in terms of the perimeter?


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• As the number of sides of the regular polygon increases, can the perimeter of the polygon ever be greater than or equal to the circumference of the inscribed circle? Explain.


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Problem 7 Getting It Together!

Recall that the area formula for a circle is A 5 πr2, where r is the radius of the circle.

Recall that the circumference formula for a circle is C 5 2πr, where

r is the radius of the circle.

1. Use a centimeter ruler to measure the radius and sides of the regular polygons in

Problems 2 through 5 and the circumference formula to complete the table.

Use 3.14 for pi.

Regular Polygon/Inscribed Circle

Side Length (s)

Perimeter (P)

Radius (r)

Circumference (C)

Equilateral Triangle

Square

Regular Pentagon

Regular Hexagon

2. What is the relationship between the perimeter of the regular polygon and the

circumference of the inscribed circle as the number of sides of the regular

polygon increases?

As the number of sides of the regular polygon increases, the perimeter of the

regular polygon gets closer to the circumference of the inscribed circle.

3. If the regular polygon had an infinitely large number of sides, how would you

describe the perimeter of the regular polygon in relation to the circumference of the

inscribed circle?

The perimeter of the regular polygon would be the same as the circumference of

the inscribed circle.

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Problem 7Students use a centimeter ruler to determine the length of the radii and sides of the regular polygons in Problems 2 through 5. They record the data in a table and use the formula C 5 2πr to calculate the circumference of each figure. They will then compute the area of each regular polygon, the area of the inscribed circle, and the area of the shaded region for each figure and record that data in a table. Students conclude that as the number of sides of the regular figures increase, the perimeter of the polygon approaches the circumference of the inscribed circle and the area of the regular polygon approaches the area of the inscribed circle.

MaterialsRuler



• How did you determine the perimeter of each regular polygon?

• How did you determine the circumference of each regular polygon?

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• How did you determine the area of each regular polygon?

• How did you determine the area of the inscribed circles?

• How did you determine the area of the shaded regions?

• As the number of sides of the regular polygon increases, can the area of the polygon ever be equal to or greater than the area of the inscribed circle? Explain.

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4. Use the measurements from Question 1 and the area formula to complete the table.

Use 3.14 for pi.

Regular Polygon/ Inscribed Circle

Area of the Regular Polygon

Area of the Inscribed Circle

Area of the Shaded Region

Equilateral Triangle

Square

Regular Pentagon

Regular Hexagon

5. As the number of sides of the regular polygon increases, what do you notice about

the area of the shaded region?

As the number of sides of the regular polygon increases, the area of the shaded

region decreases.

6. What is the relationship between the area of the regular polygon and the area of the

inscribed circle as the number of sides of the regular polygon increases?

As the number of sides of the regular polygon increases, the area of the regular

polygon gets closer to the area of the inscribed circle.

7. If the regular polygon had an infinitely large number of sides, how would you describe

the area of the regular polygon in relation to the area of the inscribed circle?

The area of the regular polygon would be the same as the area of the

inscribed circle.

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Talk the TalkStudents simplify the formula C 5 2πr until they create the formula A 5 πr2. The relationship between these two formulas is discussed.

GroupingHave students complete the Talk the Talk with a partner. Then share responses as a class.


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Talk the Talk

You have established that the area of a regular polygon is half its perimeter times the

radius of the inscribed circle or A 5 1 __ 2

Pr in Problems 2 through 6.

You have observed that as the number of sides of the regular polygon increases, it

approaches the shape of a circle in Problem 7.

Therefore, the formula A 5 1 __ 2

Pr also applies to circles.

Technically, the perimeter of a circle is the same as the circumference of a circle.

Let’s rephrase that: the area of a circle can be calculated using the formula A 5 1 __ 2

Cr.

You know that the circumference formula is C 5 2πr.

Substitute an equivalent expression for C into the area formula and simplify. What do

you notice?

A 5 1 __ 2

Cr

C 5 2πr

A 5 1 __ 2

Cr

A 5 1 __ 2

(2πr)r

A 5 πr2


AHA! Do you see how it works?

Think about all the relationships

between these two formulas.

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Have students work collaboratively on the Talk the Talk. Prompt students to read the problem together. Remind them that if they do not know an exact word, they can use ask a peer or raise their hand for help. Look for students to discuss a strategy and explain their reasoning.

Have students work

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Follow Up






Check for Students’ UnderstandingWhich pipe configuration will deliver more water to the residents, one 8 cm pipe or two 4 cm pipes?

8 cm

4 cm

4 cm

A 5 πr2

A 5 π(4)2

A 5 (3.14)(16)

A 5 50.24

2(50.24) 5 100.48

A 5 πr2

A 5 π(8)2

A 5 (3.14)(64)

A 5 200.96

The 8 cm pipe will deliver more water to the residents because the area of the pipe is

approximately 200.96 square centimeters and the total area of the two 4 cm pipes is

100.48 square centimeters.

9.3 Area of a Circle • 504A

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9.4 Unknown Measurements • 505A


Use the area and circumference formulas to solve for unknown measurements.

Use composite figures to solve for unknown measurements.

Choose the circumference or area formula based on a problem situation.

Essential Ideas

• The formula to calculate the area of a circle is A 5 πr 2.

• The formula to calculate the circumference of a circle is C 5 2πr.

• Composite figures that include circles are used to solve for unknowns.

• When solving problems involving circles, remember that the circumference formula is used to determine the distance around a circle, while the area formula is used to determine the amount of space contained inside a circle.

It’s About Circles!Unknown Measurements




(C) determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles

MaterialsCompass

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OverviewStudents use the area of a circle formula and the circumference formula to solve for unknown

measurements in problem situations. Some of the situations are problems composed of more than

one figure, and some of the situations include shaded and non-shaded regions. Students will also

determine whether to use the circumference or area formula to solve problems involving circles.

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Warm Up

1. The circumference of a circle is approximately 157 centimeters. Describe a strategy to solve for

the area of the circle.

I can use the circumference formula C 5 2πr to solve for the radius of the circle. Once I know

the radius of the circle, I can use the area formula A 5 πr2 to solve for the area of the circle.

2. Solve for the area of the circle in Question 1.

C 5 2πr

157 5 2(3.14)(r)

157 5 6.28r

r 5 25

A 5 πr2

A 5 (3.14)(25)2

A 5 1962.5

The area of the circle is approximately 1962.5 square centimeters.

9.4 Unknown Measurements • 505C

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9.4 Unknown Measurements • 505


Use the area and circumference formulas to solve for unknown measurements.

Use composite figures to solve for unknown measurements.

Choose the circumference or area formula based on a problem situation.

It’sAboutCircles!Unknown Measurements

People have book circles—groups that meet to discuss books. There are circles

of friends. There’s the circle of life. Vicious circles. Come full circle. Going around

in circles. Circle the wagons.

Are there any other common phrases you can think of that use the word circle?

Do you know what the phrase “circle the wagons” means?

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Problem 1 Fencing

A friend gave you 120 feet of fencing. You decide to fence in a portion of the backyard for

your dog. You want to maximize the amount of fenced land.

Draw a diagram, label the dimensions, and compute the maximum fenced area.

Assume the fence is free-standing and you are not using any existing structure.

C 5 2πr

120 ¯ 2(3.14)(r)

120 ¯ 6.28(r)

19.1 ¯ r

The fenced area will look like a circle with a radius of

approximately 19.1 feet.

19.1'

A 5 πr2

A ¯ (3.14) (19.1)2

A ¯ 1145.5 sq ft

A circle with a 19.1 ft radius has an area of approximately

1145.5 sq ft.

Think about circles!

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• How can you get the information needed to determine the area of a circular dog pen?

Problem 1Students are given 120 feet of fencing and asked to construct a free standing dog pen in such a way that the maximum amount of area is fenced in. The circumference formula is used to determine the radius and then the radius can be used to determine the maximum area.

MaterialsCompass

GroupingHave students complete Problem 1 with a partner. Then share the responses as a class.

Share Phase, Problem 1

• What does 120 feet of fencing represent, in terms of the shape of the dog pen?

• What shape should be used to maximize the area of the dog pen?

• If you build the pen in the shape of a square, what will be length of each side?

• What is the area of a square dog pen?

• If you build the pen in the shape of a circle, what will be the circumference of the circle?

• What information is needed to determine the area of a circular dog pen?

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Problem 2 Composite Figures

Many geometric figures are composed of two or more geometric shapes. These figures

are known as composite figures. When solving problems involving composite figures, it is

often necessary to calculate the area of each figure and then add these areas together.

1. A figure is composed of a rectangle and two semicircles. Determine the area

of the figure.

6.5 cm

13 cm

Area of the rectangle:

A 5 bh

A 5 13(6.5) 5 84.5 sq cm

Area of 1 semicircle:

A 5 1 __ 2

πr2

A 5 1 __ 2

π(3.252) 5 1 __ 2

π(10.5625) 5 5.28125π ¯ 16.583125 sq cm

Area of the figure:

A ¯ 84.5 1 2(16.583125)

A ¯ 84.5 1 33.16625 ¯ 117.66625 sq cm

If you need to round then your

area will be an approximation. Be sure

to use the correct symbols.

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Problem 2A circle is inscribed in a square and students determine the area of the shaded region by subtracting the area of the circle from the area of the square. In the second problem, two small circles are inscribed in a larger circle and the shaded region is computed two different ways. Students compare the methods of solution and note the similarities and differences. They will then practice calculating the area of the shaded region in composite figures.



• How do you determine the area of the rectangle?

• How do you determine the area of each semicircle?

• What strategy will you use to determine the area of the shaded region?


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2. A circle is inscribed in a square. Determine the area of the shaded region.

15 m

Area of the square Area of the circle

A 5 s2 A 5 πr2

A 5 (302) 5 900 sq m A 5 (π)(152) 5 225π < (3.14)(225) < 706.5

Area of the shaded region: 900 2 706.5 < 193.5 sq m

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• What shapes are in this figure?

• How do you determine the area of the circle?

• How do you determine the area of the square?

• What strategy can be used to determine the area of the shaded region?

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3. Two small circles are drawn that touch each other, and both circles touch the large

circle. Determine the area of the shaded region.

8 in. 8 in.

MatthewArea of 1 small circle

A = π(8)2

A = 64 π

Area of 2 circles

A 5 2(64π )

A 5 128π

Area of large circle

A 5 π162

A 5 256π

256 π 2 128π 5 128π

A ¯ 128(3.14)

A ¯ 401.92

This means the area of the

shaded region is about

402 sq in.

JimmyArea of 1 small circle

A ¯ 3.14(8)2

A ¯ 3.14(64)

A ¯ 200.96

Area of 2 small circles

A ¯ 2(200.96)

A ¯ 401.92

Area of large circle

A ¯ (3.14)(16)2

A ¯ (3.14)(256)

A ¯ 803.84

803.84 2 401.92 ¯ 401.92

The area of the shaded

region is about 402 sq in.

Jimmy and Matthew each said the area of the shaded region is about 402 sq in. Compare

their strategies.

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GroupingHave students complete Question 3 with a partner. Then share the responses as a class.


Group intermediate English Language Learners in pairs. Students can then collaborate to discuss the work of Jimmy and Matthew before answering parts (a) and (b). While students are in pairs, remind them to support one in other using math vocabulary accurately or if a peer is struggling to name a word.

Group intermediate English

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a. What did Jimmy and Matthew do the same?

Both Jimmy and Matthew found the area of 1 small circle first, then doubled it

to find the area of 2 small circles. They both found the area of the large circle

next. Finally, they both subtracted the area of the 2 small circles from the area

of the large circle.

b. What was different about their strategies?

Jimmy used 3.14 in place of pi throughout his equations. Matthew used π

throughout his equations until the end when he replaced π with 3.14.

c. Which strategy do you prefer?

Answers will vary.

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• Are Jimmy and Matthew both correct?

• If the two circles inside the large circle were different sizes, would Jimmy’s method still work? Explain.

• If the two circles inside the large circle were different sizes, would Matthew’s method still work? Explain.

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4. A figure is composed of a trapezoid and a semicircle. Determine the area of the figure.

7 ft

16 ft

7 ft

Area of the trapezoid:

A 5 1 __ 2

(b1 1 b2)h

A 5 1 __ 2

(16 1 7)(7) 5 1 __ 2

(23)(7) 5 80.5 sq ft

Area of the semicircle:

A 5 1 __ 2

πr2

A 5 1 __ 2

π (3.52) 5 1 __ 2

π (12.25) 5 6.125π ¯ 19.2325 sq ft

Area of the figure:

A ¯ 80.5 1 19.2325

A ¯ 99.7325 sq ft

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• How do you determine the area of the trapezoid?

• How do you determine the area of the semicircle?


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5. One medium circle and one small circle touch each other, and each circle touches the

large circle. Determine the area of the shaded region.

12 m

3 m

Area of the smallest circle

A 5 πr2

A 5 (π)(32) 5 9π

Area of the medium circle

A 5 πr2

A 5 (π)(122) 5 144π

Area of the largest circle

A 5 πr2

A 5 (π)(302) 5 900π

Area of the shaded region: 900π 2 9π 2 144π 5 747π < 2345.58 sq m

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• Could Question 5 be solved using Jimmy’s method from Question 3? Explain.

• Could Question 5 be solved using Matthew’s method from Question 3? Explain.


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6. A rectangle is inscribed in a circle. Determine the area of the shaded region.

6 cm

8 cm

10 cm

Area of the circle

A 5 πr2

A 5 (π)(52) 5 25π < 78.5 sq cm

Area of the rectangle

A 5 bh

A 5 (6)(8) 5 48 sq cm

Area of the shaded region: 78.5 2 48 < 30.5 sq cm

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• What does the dotted line represent with respect to the circle?

• What does the dotted line represent with respect to the rectangle?

• What does the dotted line represent with respect to the right triangle?

• How do you determine the area of the rectangle?


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• How do you determine the area of the regular hexagon?



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7. A circle is inside a regular hexagon. Determine the area of the shaded region.

2 in.

2 in.

6 in.

Area of the regular hexagon

A 5 1 __ 2

(P)(a)

A 5 1 __ 2

(36)(4) 5 72 sq in.

Area of the circle

A 5 πr2

A 5 (π)(22) 5 4π < (3.14)(4) < 12.56 sq in.

Area of the shaded region: 72 2 12.56 < 59.44 sq in.

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• How do you determine the area of each shape that makes up the figure?


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8. A figure is composed of a triangle and three semicircles. Determine the area

of the figure.

20 in.

20 in.20 in.

17.3 in.

Area of the triangle:

A 5 1 __ 2

bh

A 5 1 __ 2

(20)(17.3) 5 173 sq in.

Area of 1 semicircle:

A 5 1 __ 2

πr2

A 5 1 __ 2

π (102) 5 1 __ 2

π (100) 5 50π ¯ 157 sq in.

Area of the figure:

A ¯ 173 1 3(157)

A ¯ 173 1 471 ¯ 644 sq in.

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• How is this question different from the previous questions?




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9. Determine the area of the shaded region. All circles have the same radius

of 10 inches.

10 in.

Area of square

A 5 (lw)

A 5 (20 3 20)

A 5 400 sq in.

Area of 1 circle

A 5 πr2

A < (3.14)(10)2

A < (3.14)(100)

A < 314 sq in.

Area of shaded region

A < 400 2 314

A < 86 square inches

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Problem 3Students choose whether to use the circumference or area formula for real world problems involving circles.



• What words give you clues to which formula you should use?

• Why are different units used for circumference and area?

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Problem 3 Circumference or Area?

Remember that the circumference of a circle is the distance around the circle, while the

area of a circle is the amount of space contained inside the circle. When solving problems

involving circles, it is important to think about what you are trying to determine before you

decide which formula to use.

1. A city park has a large circular garden with a path around it. The diameter of the

garden is 60 feet.

a. Gina likes to walk along the circular path during her lunch breaks. How far does

Gina walk if she completes one rotation around the path?

I want to know the distance around the circle, so I will use the formula for the


C 5 2πr

C 5 2π ( 60 ___ 2

) C ¯ 2(3.14)(30)

C ¯ 188.4

Gina walks about 188.4 feet if she completes one rotation around the path.

b. Jason works for the City Park Department. He needs to spread plant food all over

the garden. How much of the park will he cover with plant food?

I want to know the amount of space inside the circle, so I will use the formula

for the area of a circle.

A 5 πr2

A 5 π ( 60 ___ 2

) 2 A ¯ (3.14)(3 0 2 )

A ¯ 2826

Jason will cover about 2826 square feet with plant food.


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2. Samantha is making a vegetable pizza. First, she presses the dough so that it fills a

circular pan with a 16-inch diameter. The next step is for her to cover it with sauce.

How much of the pizza will she cover with sauce?

I want to know the amount of space inside the circle, so I will use the formula for

the area of a circle.

A 5 πr2

A 5 π ( 16 ___ 2

) 2 A ¯ (3.14)( 8 2 )

A ¯ 200.96

Samantha will cover about 200.96 square inches with sauce.

3. Members of a community center have decided to paint a large circular mural in the

middle of the parking lot. The radius of the mural is to be 11 yards. Before they begin

painting the mural, they use rope to form the outline. How much rope will they need?

I want to know the distance around the circle, so I will use the formula for the


C 5 2πr

C 5 2π (11)

C ¯ 2(3.14)(11)

C ¯ 69.08

They will need about 69.08 feet of rope.


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Follow Up






Check for Students’ UnderstandingThe radius of the small circle is 0.5 millimeters. The area of the large circle is 28.26 square millimeters.

What is the area of the shaded region? (Use 3.14 for π)

A 5 πr2

A 5 (3.14)(.5)2

A 5 0.785

28.26 2 0.785 5 27.475

The approximate area of the shaded region is 27.475 millimeters.

9.4 Unknown Measurements • 518A

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Chapter 9 Summary • 519

Defining a Circle and Its Properties

A circle is a collection of points on the same plane equidistant from the same point. The

center of a circle is the point from which all points on the circle are equidistant. Circles are

named by their center point. The radius of a circle is a line segment formed by connecting

a point on the circle and the center of the circle. The diameter of a circle is a line segment

formed by connecting two points on the circle such that the line segment passes through

the center point.

Example

Circle Z is shown. Point Z is the center of the circle. Line segment AZ is a radius of

Circle Z. Line segment XY is a diameter of Circle Z.

Y

A

XZ

Chapter 9 Summary

Key Terms circle (9.1)

center of a circle (9.1)

radius of a circle (9.1)

diameter of a circle (9.1)

pi (9.2)

concentric circles (9.3)

annulus (9.3)

inscribed circle (9.3)

I like geometry more than algebra. But now I see how

they're kind of connected.


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Exploring the Relationship between the Diameter and Circumference of a Circle

Recall that the circumference of a circle is the distance around the circle. The number

pi (π) is the ratio of the circumference of a circle to its diameter. That is,

pi 5 circumference of a circle _______________________ diameter of a circle

or π 5 C __ d

, where C is the circumference of the circle

and d is the diameter of the circle.

Example

The diameter of Circle M is 10 cm. The circumference of Circle M is 31.4 cm.


M 5 cm 10 cm 31.4 cm 31.4 ____ 10

5 3.14

Calculating the Circumference of a Circle

The circumference of a circle can be calculated using the formula C 5 πd or the formula

C 5 2πr, where C represents the circumference of the circle, d represents the diameter of

the circle, and r represents the radius of the circle. The value for π is often rounded to

3.14. The formula can also be used to calculate the diameter or radius of a circle when the

circumference is known.

Example

The diameter of a circle is 54 cm. To calculate the circumference, use the formula C 5 πd.

C 5 πd

< 3.14(54)

< 169.56

The circumference of the circle is approximately 169.56 cm.


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Using Models to Approximate the Area of a Circle

Approximate the area of a circle by dividing the circle into equal parts and then flipping

every other part vertically and placing them side by side. This new figure will closely

resemble a parallelogram, and the area of a parallelogram is equal to the product of the

base and height.

Example

Eight diameters divide a circle into 16 equal parts, as shown in Figure 1. Figure 2 show

the 16 parts separated. If you flip every other part vertically and place all 16 parts

side-by-side, it will look like Figure 3.

Figure 3

Figure 1 Figure 2

Figure 3 most closely resembles a parallelogram. The length of the base b is

approximately equal to half of the circumference of the circle C, or b 1 __ 2 C. The height h is

approximately equal to the radius of the circle r, or h r. So, the area of the circle can be

approximated by the formula A bh 1 __ 2

C ? r.

Determining the Area of a Circle

The area of a circle is the amount of space inside a circle. Determine the area of a circle A

by using the formula A 5 πr2 where r represents the radius.

Example

A circle has a radius of 7 feet.

A 5 πr2

A < 3.14(7)2

A < 153.86

The area of the circle is approximately 153.86 square feet.


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Determining the Area of an Annulus

Concentric circles are circles that share the same center. An annulus is the region

bounded by two concentric circles.

Example

8 cm

6 cm

Area of the larger circle: Area of the smaller circle:

A 5 πR2 A 5 πr2

A 5 π(8)2 A 5 π(6)2

A 5 64π A 5 36π

Area of the annulus 5 64π 2 36π 5 28π < 87.92

The area of the annulus is approximately 87.92 square centimeters.


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Using Circles Inscribed in Regular Polygons to Explore Area and Perimeter

An inscribed circle is a circle that fits exactly within the boundaries of another shape. It is

the largest possible circle that will fit inside a plane figure. As the number of sides of the

regular polygon increases, the perimeter of the regular polygon gets closer to the

circumference of the inscribed circle. As the number of sides of the regular polygon

increases, the area of the regular polygon gets closer to the area of the inscribed circle.

The area of a regular polygon is half its perimeter times the radius of the inscribed circle.

Example

The radius of Circle T is equal to the height of each of the eight triangles in the octagon.

S

S

T

r

S

SS

S

S

S

The area of each triangle is A 5 1 __ 2

sr.

The area of the octagon is A 5 1 __ 2

sr 3 8.

The perimeter of the octagon is P 5 8s.

The area of the octagon in terms of its perimeter is A 5 1 __ 2

Pr.


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Exploring the Relationship between the Circumference and Area of a Circle

The formula for the area of a regular polygon is A 5 1 __ 2

Pr, where P is the perimeter of the

polygon and r is the radius of an inscribed circle. As the number of sides of a regular

polygon increases, the shape of the polygon approaches the shape of a circle. Therefore,

the formula A 5 1 __ 2

Pr can also be applied to circles.

Example

The perimeter or circumference of a circle can be calculated using the formula C 5 2πr.

By inserting the expression for the perimeter of a circle into the equation A 5 1 __ 2 Pr, it is

determined that the area of a circle is A 5 1 __ 2

(2πr)r or A 5 πr2.

Using Area and Circumference Formulas to Solve for Unknown Measurements

When solving problems involving circles with unknown measurements, the area and

circumference formulas can be used to determine the measurements. A problem may

require the use of both formulas to determine the answer.

Example

To calculate the area of a circle with a circumference of 15.7 meters, first determine the

radius of the circle using the circumference formula.

C 5 2πr

15.7 < 2(3.14)r15.7 < 6.28r

15.7 _____ 6.28

< 6.28r _____ 6.28

2.5 m < r

The radius of the circle is 2.5 meters. Calculate the area of the circle using the

area formula.

A 5 πr2

A < π(2.5)2

A < π(6.25)

A < 19.625 m2

A circle with a circumference of 15.7 meters has an area of approximately

19.625 square meters.


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ngDetermining the Area of Composite Figures

Many geometric figures are composed of two or more geometric shapes. When solving

problems involving composite figures, it is often necessary to calculate the area of each

figure, and then add these areas together.

Example

A composite figure is made from a rectangle, two semicircles, and a triangle.

8 cm

6 cm4 cm

Area of the rectangle: Area of one semicircle: Area of the triangle:

A 5 bh A 5 1 __ 2

πr2 A 5 1 __ 2 bh

A 5 (8)(6) 5 48 sq cm A 5 1 __ 2

π(4)2 5 8π < 25.12 sq cm A 5 1 __ 2 (6)(4) 5 12 sq cm

Area of the figure < 48 1 2(25.12) 1 12 < 110.24 square centimeters



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Using Composite Figures to Solve for Unknown Measurements

Many geometric figures are composites of two or more geometric shapes. When solving

problems involving composite figures, it is often necessary to calculate the area of each

geometric shape which composes the figure.

Example

A circle with a radius of 9 inches is inscribed inside a regular pentagon with side lengths of

13.1 inches. To calculate the area of the shaded region, subtract the area of the circle from

the area of the pentagon.

9 in. 13.1 in.

Calculate the area of the pentagon using the formula A 5 1 __ 2

Pr. The perimeter of the

pentagon is 5(13.1) or 65.5 inches.

A 5 1 __ 2

Pr

A 5 1 __ 2 (65.5)(9)

A 5 294.75 in.2

Calculate the area of the circle.

A 5 πr2

A < (3.14)(9)2

A < (3.14)(81)

A < 254.34 in.2

The area of the shaded region is approximately 294.75 2 254.34, or 40.41 square inches.


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Choosing the Circumference or Area Formula Based on a Problem Situation

To determine when to use the circumference formula or the area formula, first analyze the

problem situation. If the problem situation refers to the distance around the outside of a

circle, then the circumference formula is needed. If the problem situation refers to covering

a circle, then the area formula is needed.

Example

The town of Bridgeville is building a circular race car track. The track will have a radius of

60 yards.

To determine the distance traveled in one rotation, use the circumference formula.

C 5 2πr

C 5 2π(60) 5 120π < 376.8

The distance traveled in one rotation is approximately 376.8 yards.

To determine the amount of space the track covers, use the area formula.

A 5 πr2

A 5 π(60)2 5 3600π < 11,304

The track covers approximately 11,304 square yards.



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Documents

Circles - Austin Independent School Districtcurriculum.austinisd.org/.../M_7_Carnegie_Circles.pdf · to the length of its diameter. X 9.3 Area of a Circle 7.8.B 7.8.C 1 ... Step 2:center