Area of a Circle - Everyday Math · PDF filethe area of a circle. ... use to find the circumference of the circle. b. The circumference of the circle is about centimeters. 4

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

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AssessmentManagement

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Lesson 10�9 831

Advance PreparationFor Part 1, copy the table on journal page 365 onto the board. Make transparencies of Math Masters,

pages 314 and 436.

Teacher’s Reference Manual, Grades 4–6 pp. 64– 68, 185, 186, 221

Key Concepts and Skills• Find the median of a data set.

[Data and Chance Goal 2]

• Investigate and apply a formula for finding

the area of a circle.

[Measurement and Reference Frames Goal 2]

• Use ratios to describe the relationship

between radius and area.

[Measurement and Reference Frames Goal 2]

• Use patterns in a table to define the

relationship between radius and area.

[Patterns, Functions, and Algebra Goal 1]

Key ActivitiesStudents draw circles by tracing round

objects on centimeter grids. They measure

the areas and radii and find that the ratio of

a circle’s area to the square of its radius

is close to the value of π. They use the

formula to calculate the areas of the circles.

Ongoing Assessment: Informing Instruction See page 833.

MaterialsMath Journal 2, pp. 364–366B

Study Link 10�8

Math Masters, p. 436

transparencies of Math Masters, pp. 314 and

436 � slate � collection of round objects from

Lesson 10�8 � calculator � metric ruler

Converting Units of MeasureMath Journal 2, pp. 366A and 366B

Students convert units of measure.

Playing First to 100Student Reference Book, p. 308

Math Masters, pp. 456–458

per partnership: 2 six-sided dice,

calculator

Students practice solving open

number sentences.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 458. [Patterns, Functions, and Algebra Goal 2]

Math Boxes 10�9Math Journal 2, p. 367

Students practice and maintain skills

through Math Box problems.

Study Link 10�9Math Masters, p. 315

Students practice and maintain skills

through Study Link activities.

ENRICHMENTModeling πr 2

Math Masters, pp. 316 and 317

scissors � colored pencil or marker �

construction paper � glue or tape

Students apply their understanding of area

formulas to verify the formula for the area

of a circle.

EXTRA PRACTICE

Calculating the Circumferences and Areas of CirclesMath Masters, p. 318

Geometry Template � calculator

Students use formulas to solve problems

involving circumferences and areas of circles.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

� Area of a CircleObjective To introduce a formula to calculate the area

of a circle.o

Common Core State Standards

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Math Message

Use the circle at the right to solve Problems 1�4.

1. The diameter of the circle is

about centimeters.

2. The radius of the circle is

about centimeters.

3. a. Write the open number sentence you would

use to find the circumference of the circle.

b. The circumference of the circle is

about centimeters.

4. Find the area of this circle by counting squares.

About cm2

5. What is the median of all the

area measurements in your class? cm2

6. Pi is the ratio of the circumference to the diameter of a circle. It is also

the ratio of the area of a circle to the square of its radius. Write the

formulas to find the circumference and the diameter of a circle that use

these ratios.

The formula for the circumference of a circle is .

The formula for the area of a circle is .

50

25

4

8

Answers vary.

Measuring the Area of a CircleLESSON

10 �9

Date Time

1cm2

C � π � 8, or C � 3.14 � 8

C � π � d, or C � πdA � π � r2, or A � πr2

Math Journal 2, p. 364

Student Page

832 Unit 10 Using Data; Algebra Concepts and Skills

Getting Started

Math MessageSolve Problems 1–4 on journal page 364.

Study Link 10�8 Follow-UpHave partners compare answers and resolve differences.

Mental Math and Reflexes Have students record an appropriate unit of measure for given situations. Suggestions:

Amount of carpet needed to carpet a bedroom ft2, yd2, or m2

Distance from where you live to a summer camp on a lake Miles, or kilometers

Length of a dollar bill Inches, or centimeters

Amount of juice the average person drinks in a week Cups, gallons, or liters

Volume of cubes that could be stacked in a desk drawer, filling every space cm3 or in3

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 364; Math Masters, p. 314)

Verify that students were able to determine the diameter, radius, and circumference of the circle. Then ask students to share their answers to Problem 4. List their measurements in order on the board or a transparency, find the median of the measurements, and have students record the median in Problem 5 on the journal page.

Point out the wide variation in the area measurements. Discuss why it is difficult to measure the area of a circle by counting squares. The pieces are irregular. Then use the transparency of Math Masters, page 314 to demonstrate the following method for counting squares.

1. Make a check mark in each whole centimeter square.

2. Mark each centimeter square that is nearly a whole square with an X.

3. Find combinations of partial squares that are about equivalent to a whole square. Number each set, using the same number in each partial square.

4. Count the approximate total number of squares. In the circle in Problem 1, the area of the circle is about 52 square centimeters.

Ask the class to suggest situations in which one might need to find the area of a circle. To support English language learners, list students’ suggestions on the board. Explain that students will count squares and use a formula to find the area of circles.

ELL

NOTE All measurement is inexact, but

finding the area of a circle by counting square

centimeters is especially so. Area, unlike

length, volume, or mass, is difficult to measure

directly. Area is usually found by measuring

lengths and applying a formula. Nevertheless,

if students are particularly precise, measuring

area by counting square units can be an

accurate and practical technique.

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Areas of CirclesLESSON

10 �9

Date Time

Work with a partner. Use the same objects, but make separate measurements so you

can check each other’s work.

1. Trace several round objects onto the grid on Math Masters, page 436.

2. Count square centimeters to find the area of each circle.

3. Use a ruler to find the radius of each object. (Reminder: The radius is half the

diameter.) Record your data in the first three columns of the table below.

4. Find the ratio of the area to the square of the radius for each circle. Write the ratio

as a fraction in the fourth column of the table. Then use a calculator to compute the

ratio as a decimal. Round your answer to two decimal places, and write it in the

last column.

5. Find the median of the ratios in the last column.

Answers vary.

Ratio of Area to Radius Squared

ObjectArea Radius

(sq cm) (cm) as a Fraction ��rA2�� as a Decimal


Student Page

Lesson 10�9 833

▶ Exploring the Relationship

WHOLE-CLASS ACTIVITY

between Radius and Area(Math Journal 2, pp. 364 and 365;

Math Masters, p. 436)

Algebraic Thinking This exploration requires students to measure the radius of circles where the center is not given and to find the approximate areas of these circles. Demonstrate what students are to do using a round object and the transparency of Math Masters, page 436:

1. Make a circle on the transparency by tracing the object.

2. Find the approximate area of the circle by counting squares. Record the name of the object and its approximate area in the first and second columns of the table on the board.

3. Use a right-angled corner of a piece of paper to find a diameter of the circle. Position the right angle on the circle as shown below. Mark the points where the sides of the angle intersect the circle. These are the endpoints of a diameter of the circle. Measure the distance between endpoints.

diam

eter

4. Ask students how to find the radius when the diameter is known. Divide by 2. Record the radius in the third column of the table.

Ongoing Assessment: Informing Instruction

Watch for students who struggle with measuring accurately. Have them trace

and measure larger objects. Measurement errors are less significant with

longer lengths.

Have partners trace several round objects on a centimeter grid using Math Masters, page 436. Then have students measure the radius and the area of each of the tracings using the techniques just demonstrated. Partners use the same objects but measure independently and check each other’s work. They record their results in the first three columns of the table on the journal page.

When all groups have traced and measured at least three objects, bring the class together to demonstrate how to complete the last two columns in the table on the journal page. Continue to use the circle you traced on the transparency earlier.

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BBBBBBBBBBBBBBBBBBBB EEELELEMMMMMMMMMOOOOOOOOOBBBLBLBLBLBLBBBOOORORORORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLLVINVINVINNNNVINVINVINVINVINVINVINVINV GGGGGGGGGGGOOLOOLOLOLOLOOLOO VINVVINLLLLLLLLLLVINVINVINVINVINNVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLLOOO VVVLLLLLLLLLLVVVVVVVVOSOSOSOOSOSOSOSOSOSOOSOSOSOOSOOOOSOSOSOSOSOSOSOOOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLVVVVVVVVLLLLLLLVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING


A Formula for the Area of a CircleLESSON

10 �9

Date Time

Your class just measured the area and the radius of many circles and found

that the ratio of the area to the square of the radius is about 3.

This was no coincidence. Mathematicians proved long ago that the ratio

of the area of a circle to the square of its radius is always equal to π.

This can be written as:

A

_ r 2

= π

Usually this fact is written in a slightly different form, as a formula for

the area of a circle.

1. What is the radius of the circle in the Math Message on journal page 364? 4 cm

2. Use the formula above to calculate the area of that circle. 50.27 cm2

3. Is the area you found by counting square centimeters

more or less than the area you found by using the formula?

How much more or less?

4. Use the formula to find the areas of the circles you traced on Math Masters, page 436.

5. Which do you think is a more accurate way to find the area of a circle, by counting

squares or by measuring the radius and using the formula? Explain.

The formula for the area of a circle is

A = π ∗ r 2

where A is the area of a circle and r is its radius.

The formula is more accurate because it tellsexactly how the area and radius are related.Counting squares is difficult and less accuratebecause partial squares are irregular.

Sample answers:

78.54 cm2 176.71 cm2 28.27 cm2

Answers vary.

Answers vary.

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Student Page

Converting Units of MeasureLESSON

10�9

Date Time

1. Tell if you should multiply or divide.

a. To convert from a larger unit to a b. To convert from a smaller unit to a

smaller unit (such as from ft to in.), larger unit (such as from m to km),

you

. you

.

2. Find the equivalent customary measurement.

a. 24 in. =

ft b. 3 yd =

in.

c. 12 qt =

gal d. 3.5 gal =

qt

e. 1.5 mi =

ft =

yd f. 2.5 qt =

pt =

c

3. Find the equivalent metric measurement.

a. 10 m =

cm b. 50 mm =

cm

c. 250 g =

kg d. 2.5 kL =

L

e. 7.5 m =

cm =

mm f. 100 mg =

g =

kg

Metric System

Length Mass Liquid Capacity

1 centimeter (cm) = 1 gram (g) = 1 liter (L) =

10 millimeters (mm) 1,000 milligrams (mg) 1,000 milliliters (mL)

1 meter (m) = 1 kilogram (kg) = 1 kiloliter (kL) =

100 centimeters (cm) 1,000 grams (g) 1,000 liters (L)

1 kilometer (km) =

1,000 meters (m)

Customary System

Length Weight Liquid Capacity

1 foot (ft) = 12 inches (in.) 1 pound (lb) = 16 ounces (oz) 1 pint (pt) = 2 cups (c)

1 yard (yd) = 3 feet (ft) 1 ton (T) = 2,000 pounds (lb) 1 quart (qt) = 2 pints (pt)

1 mile (mi) = 5,280 feet (ft) 1 gallon (gal) = 4 quarts (qt)

multiply divide

2

1,000

3

0.25

7,920

750 0.1

2,640

7,500 0.0001

108

5

14

2,500

5 10

366A-366B_EMCS_S_MJ2_G5_U10_576434.indd 366A 3/22/11 12:43 PM

Math Journal 2, p. 366A

Student Page


� In the fourth column of the table on the board, write the ratio of the circle’s area to the radius squared as a fraction. Ask volunteers for another way to express the meaning of radius squared. radius ∗ radius

� Use a calculator to convert the fraction to a decimal, rounded to two decimal places. Write the resulting decimal in the fifth column of the table.

Have students complete the last two columns of the table on the journal page, find the median of the ratios in the last column, and record it as the answer to Problem 5.

When students have finished, ask them to share the values they found and record them on the board. Most median values should be close to 3, though some might be far off because of various errors—using the diameter instead of the radius, for example, or measuring the radius in inches rather than centimeters.

Ask what number the ratios are close to. π Explain that this is no coincidence: The ratio of the area of a circle to the square of its radius is always equal to π. Help students recognize how remarkable this is—the same number, π, is the ratio of the circumference to the diameter and the ratio of the area to the radius squared. These ratios are the bases for the formulas that can be used to find the circumference and the area of a circle. Have students write the formulas in Problem 6 on journal page 364.

▶ Using a Formula to Find the

WHOLE-CLASS ACTIVITY

Area of a Circle(Math Journal 2, pp. 364–366; Math Masters, p. 436)

Algebraic Thinking Have students read journal page 366 and use the formula to calculate the areas of the Math Message circle and the circles they traced on Math Masters, page 436. They compare the areas they found by counting square centimeters with the areas from the formula.

2 Ongoing Learning & Practice

▶ Converting Units of Measure (Math Journal 2, pp. 366A and 366B)

Students convert different-size units of measure, compare units of measure, and solve problems involving measurement conversions.


Date Time

Converting Units of Measure continuedLESSON

10�9 4. Which is less?

a. 1.5 gallons or 20 cups b. 1.25 L or 1,300 mL

5. Which is more?

a. 1 3

_ 4 lb or 28 oz b. 1,299 g or 1.3 kg

6. Arrange each set of measurements in order from least to greatest.

a. 9 oz, 1 _ 2 lb, 0.75 lb b. 0.75 m, 800 mm, 85 cm

7. Arrange each set of measurements in order from greatest to least.

a. 0.75 yd, 1.5 ft, 39 in. b. 2,500 g, 100,000 mg, 2.55 kg

8. The standard length of a marathon is

26 miles 385 yards. How many yards

is that in all?

9. You can estimate the adult height of a 2-year-old child by doubling the child’s

height. Suppose a 2-year-old child is 35 1 _ 2 inches tall. Estimate what the child’s

adult height will be in feet and inches.

10. Mt. McKinley in Alaska is 6.194 km tall. This is 705 m greater than the height

of Mt. St. Elias in Alaska. How tall is Mt. St. Elias in kilometers?

11. A recipe for chicken soup calls for 12 cups of chicken broth.

a. How many gallons is that equivalent to?

b. How many gallons are needed if the recipe is doubled?

They are equal.

46,145 yards

5 feet 11 inches

5.489 km

1 _

2 lb, 9 oz, 0.75 lb

39 in., 0.75 yd, 1.5 ft

1.3 kg

0.75 mm, 800 mm, 85 cm

2.55 kg, 2,500 g, 100,000 mg

12

_ 16 , 3

_ 4 , or 0.75 gallon

1 1 _

2 , or 1.5 gallons

20 cups 1.25 L

366A-366B_EMCS_S_MJ2_G5_U10_576434.indd 366B 4/6/11 11:02 AM

Math Journal 2, p. 366B

Student Page

Math Boxes LESSON

10 �9

Date Time

4. Complete the “What’s My Rule?” table and

state the rule.

Rule: Subtract 12 from in,

1. Monica is y inches tall. Write an algebraic expression for the height of each person below.

a. Tyrone is 8 inches taller than Monica. Tyrone’s height:

b. Isabel is 1 1

_ 2 times as tall as Monica. Isabel’s height:

c. Chaska is 3 inches shorter than Monica. Chaska’s height:

d. Josh is 10 1

_ 2 inches taller than Monica. Josh’s height:

e. If Monica is 48 inches tall, who is the tallest person listed above? Isabel How tall is that person? 72 in.

2. Use a calculator to rename each of the

following in standard notation.

a. 217 = 131,072

b. 76 = 117,649

c. 610 = 60,466,176

d. 310 = 59,049

e. 59 = 1,953,125

3. Solve. Solution

a. – 12 + d = 14

b. 28 – e = – 2

c. b + 18 = – 24

d. – 14 = f – 7

e. 12 = 16 + g

y + 10 1

_ 2

or in - 12 in out

2

-10

0

16 4

3

-5

12

-9

7

y + 8

y - 3

1 1

_ 2 ∗ y, or 1

1

_ 2 y

218

6 92–94219

231 232 196 197

5. Find the volume of the cube.

Volume = length * width * height

Volume =

6 units

cube

216 units3

d = 26

e = 30

b = – 42

f = – 7g = – 4

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Student Page

Lesson 10�9 835

▶ Playing First to 100 PARTNER ACTIVITY

(Student Reference Book, p. 308; Math Masters,

pp. 456–458)

Algebraic Thinking Students play First to 100 to practice solving open number sentences. This game was introduced in Lesson 4-7. For detailed instructions, see Student Reference Book, page 308.

Ongoing Assessment: Math Masters

Page 458 �Recognizing Student Achievement

Use the First to 100 Record Sheet (Math Masters, page 458) to assess

students’ facilities with replacing variables and solving problems. Students are

making adequate progress if their number sentences and solutions are correct.

[Patterns, Functions, and Algebra Goal 2]

▶ Math Boxes 10�9

INDEPENDENT ACTIVITY

(Math Journal 2, p. 367)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 10-5 and 10-7. The skill in Problem 5 previews Unit 11 content.

Writing/Reasoning Have students write a response to the following: Exchange the exponent and base for each of the numbers in Problem 2, for example, 217 → 172. Write the

standard notation for each new number. 289; 279,936; 1,000,000; 1,000; 59,049 Which number is larger than its corresponding original number? Explain why. Sample answer: The original number 76 in standard notation is 117,649; 67 or 279,936 is larger than the original number because 6 used as a factor 7 times is greater than 7 used as a factor 6 times.

▶ Study Link 10�9


(Math Masters, p. 315)

Home Connection Students identify the best measurement to find in specific situations. They solve a set of problems using the formulas for area and circumference of circles.

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Teaching Master

LESSON

10�9

Name Date Time

More Area and Circumference Problems

Measure the diameter of the circle at the right to the nearest centimeter.

1. The diameter of the circle is .

2. The radius of the circle is .

3. The circumference of the circle is .

4. The area of the circle is .

5. Explain the meaning of the word circumference.

6. a. Use your Geometry Template to draw a circle that has a diameter of 2 centimeters.

b. Find the circumference of your circle.

c. Find the area of your circle.

7. a. Use your Geometry Template

to draw a circle that has

a radius of 1�1

2� inches.

b. Find the circumference of

your circle.

c. Find the area of your circle.

7.07 in.2

9.42 in.

6.28 cm

The circumference is the perimeter of the circle.

12.57 cm2

12.57 cm

2 cm

4 cm

Circle Formulas

Circumference: C = π º dArea: A = π º r 2

where C is the circumference of a circle, A is its area, d is its diameter,

and r is its radius.

Sample answer:

3.14 cm2


STUDY LINK

10�9 Area and Circumference

315

187 194

Name Date Time

Circle the best measurement for each situation described below.

1. What size hat to buy (Hint: The hat has to fit around a head.)

area circumference perimeter

2. How much frosting covers the top of a round birthday cake


3. The amount of yard that will be covered by a circular inflatable

swimming pool


4. The length of a can label when you pull it off the can


Fill in the oval next to the measurement

that best completes each statement.

5. The radius of a circle is about 4 cm. The area of the circle is about

12 cm2 39 cm2 50 cm2 25 cm2

6. The area of a circle is about 28 square inches. The diameter of the circle is about

3 in. 6 in. 9 in. 18 in.

7. The circumference of a circle is about 31.4 meters. The radius of the circle is about

3 m 5 m 10 m 15 m

8. Explain how you found your answer for Problem 7.

Sample answer: The circumference is about31.4 meters, and this equals π º d or about3.14 º d. Because 3.14 º 10 � 31.4, thediameter is about 10 meters. The radius is half the diameter or about 5 meters.

Area of a circle: A � π º r 2

Circumference of a circle: C � π º d


Study Link Master


3 Differentiation Options

ENRICHMENT PARTNER ACTIVITY

▶ Modeling πr 2 15–30 Min

(Math Masters, pp. 316 and 317)

Algebraic Thinking To apply students’ understanding of area formulas, have them cut a circle into pieces and arrange them in the shape of a parallelogram. They draw and label the parts of the parallelogram to compare them to the parts of the circle. Partners follow the directions on Math Masters, page 316 to verify the formula for the area of a circle. When students have finished, discuss any difficulties they encountered.

EXTRA PRACTICE


▶ Calculating the Circumferences 5–15 Min

and Areas of Circles(Math Masters, p. 318)

Algebraic Thinking Students use formulas to solve problems involving the circumferences and areas of circles. They draw the circles before finding the areas and circumferences. Remind students to use the fix function on their calculators to round the calculations to the nearest hundredth.

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Area of a Circle - Everyday Math · PDF filethe area of a circle. ... use to find the circumference of the circle. b. The circumference of the circle is about centimeters. 4