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Name ___________________________ Period __________ Date ___________

Geometry and Measurement Unit (Student Packet) GEO2 – SP

GEO2.1 Circumference

• Use multiple representations to explore the relationship between the diameter and the circumference of a circle.

• Understand the concept of π . • Derive the circumference formula for circles. • Solve circumference application problems.

1

GEO2.2: Area of Circles • Derive the area formula for circles. • Solve application problems that involve areas of circles.

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GEO2.3 Vocabulary, Skill Builders, and Review 14

GEO2 STUDENT PAGES

GEOMETRY AND MEASUREMENT Student Pages for Packet 2: Circles

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Circles

Geometry and Measurement Unit (Student Packet) GEO2 – SP0

WORD BANK (GEO2)

Word Definition Example or Picture

center of a circle

chord

circle

circumference

diameter

pi

radius

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Circles 2.1 Circumference


CIRCUMFERENCE

Ready (Summary)

We will explore the relationship between a circle's diameter and its circumference. We will learn about historical approximations for π . We will derive the formula for the circumference of a circle and use it to solve problems.

Set (Goals)

• Use multiple representations to explore the relationship between the diameter and the circumference of a circle.

• Understand the concept of π . • Derive the circumference formula for

circles. • Solve circumference application

problems.

Go (Warmup)

Use the figure to answer the questions below.

1. Points on a circle are all equidistant from its ___________. In the figure, this is represented by ___________.

2. A line segment from the center of a circle to any point on the circle is called a ___________. In the figure, this is represented by ___________.

3. A line segment with both endpoints on the circle is called a ___________.

In the figure, this is represented by ___________. 4. A chord that goes through the center of the circle is called a ___________.

In the figure, this is represented by ___________.

A

B C

M

D

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MEASURING CIRCLES Use the table to record the diameter and circumference of objects measured in class.

Object Diameter (d) Circumference (C) Circumference ( )

Diameter ( )C

d

A.

B.

C.

D.

E.

1. The circumference of a circle is equal to about _______ times the length of the diameter. 2. The diameter of a circle is _______ times the radius. 3. Let d represent the length of the diameter. Let r represent the length of the radius. Let C represent the length of the circumference. a. Write an equation to describe the relationship between the circumference and the

diameter. Use the symbol “≈ ” to represent “is about equal to.” b. Write an equation to describe the relationship between the diameter and the radius.

c. Write an equation to describe the relationship between the circumference and the

radius. Use the symbol “≈ ” to represent “is about equal to.”

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A LITTLE HISTORY Many civilizations over the centuries have observed that the ratio of the circumference to the diameter of a circle is constant. For example, the Romans observed that the number of paces around the outer portion of their circular temples was about three times the number of paces through the center. In mathematics, the Greek letter π (pronounced “pi”) is used to represent this ratio. There is no fraction of integers that represents the exact ratio of the circumference to the diameter, or π . Here are some approximations used by different civilizations over the ages.

Fraction used as approximation for π Decimal approximation for π (to the nearest ten-thousandth)

1. Egyptian: 25681

2. Greek: between 227

and 22371

3. Hindu: 3,9271,250

4. Roman: 377120

5. Chinese: 355113

6. Babylonian: 258

Today, the decimal approximation of π , correct to five decimal places, is

3.14159 7. Write this decimal approximation in words. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 8. Round this decimal approximation to the nearest ten-thousandth. ____________________ 9. Which civilization had the best approximation for π? ______________________________

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APPROXIMATING VALUES FOR π

There is no fraction of integers that represents the exact value of π . When solving algebra problems that involve π , it is typical to keep the symbol in the problem and solution. However, in real-life calculations, it is often necessary to approximate an answer that involves π . Compute two numerical approximations for each measurement that represents the circumference of a circle. (Hint: use your number sense to decide which approximations to compute.)

Circumference of the circle

Approximate using π ≈ 3

Approximate using π ≈ 3.14

Approximate using

π ≈227

1. π7

2. π2.8

3. π100

4. π6

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USE YOUR CIRCLE KNOWLEDGE 1

The symbol for pi is ___________. Common approximations for pi are ___________, ___________, and ____________. Formulas for the circumference of a circle are ______________ and _____________.

Draw a picture, write an appropriate formula, and substitute to solve each problem.

1. The plate problem: Calculate the circumference of a plate with a radius of 14 cm.

2. The can problem: Calculate the diameter of the top of a soup can with a circumference of 32 cm.

a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question


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USE YOUR CIRCLE KNOWLEDGE 1 (continued) Draw a picture, write the appropriate formula(s), and substitute to solve each problem.

3. The earth’s orbit problem: The earth is about 93,000,000 miles from the sun, and the earth revolves around the sun one time per year. If the earth’s orbit is approximately a circle, how far does the earth travel in one year?

4. The school track problem: A field at a local school is surrounded by a track. The straight-aways are each 425 feet. The distance across the field (top to bottom in the diagram) is 150 feet. Find the distance around the track.


a. Label the figure Hint: The field is a rectangle with half-circles (or semicircles) at both ends. b. Write the appropriate formulas c. Substitute and solve d. Answer the question

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Circles 2.2 Area of a Circle


AREA OF CIRCLES

Ready (Summary)

We will use our knowledge of the area of rectangles and circumference of circles to derive the area formula for a circle. We will use the formula to solve problems.

Set (Goals)

• Derive the area formula for circles. • Solve application problems that involve

areas of circles.

Go (Warmup)

Suppose the vertical and horizontal length between adjacent dots represents 4 feet. 1. Make a scale drawing of a 16 feet by 24 feet rectangle and find its area. 2. Make a scale drawing of a circle with a radius of 8 feet and find its circumference.

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DERIVING THE AREA OF A CIRCLE

1. What is the formula for the circumference of a circle in terms of its radius r? __________________________________________________________________ 2. Your teacher will give you a paper circle. Fold the circle in half. Fold it in half a second time.

Fold it in half a third time. 3. Unfold your circle and cut along the folds to make 8 wedges. (You may want to try more

than 8 wedges, like 16, but don’t do less than 8.) 4. Arrange the wedges in a row. Alternate the tips up and down to form a shape that

resembles a rectangle or parallelogram. Tape, glue stick, or sketch the shape here. Use your knowledge of the area of a rectangle and circumference of a circle to find the area of this shape. (Remember, it began as a circle and is becoming closer to resembling a rectangle. The more wedges you make, the closer it gets to becoming a rectangle.) 5. The width of the “rectangle” is _________________________________ of the circle.

6. The length of the “rectangle” is ________________________________ of the circle. 7. Write an equation for the approximate area of the “rectangle.” __________________________________________________________________ 8. Substitute the expression for the circumference of the circle from #1 above into your

formula and simplify. 9. What is the formula for the area of a circle, in terms of the radius? __________________________________________________________________

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USE YOUR CIRCLE KNOWLEDGE 2 Draw a picture, write an appropriate formula, and substitute to solve each problem.

1. The dish problem: Find the area of a plate whose diameter is 12 inches.

2. The water sprinkler problem: A revolving water sprinkler sprays water in a circular fashion to a distance of 20 feet in all directions. What area of grass can it cover?



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USE YOUR CIRCLE KNOWLEDGE 2 (continued) Label the pictures, write the appropriate formulas, and substitute to solve each problem.

3. The shaded area problem: The largest possible circle is to be cut from a square board that is 56 inches on each side. What is the approximate area, in square inches, of the remaining board (shaded area)?

4. The school track problem (revisited): A field at a local school is surrounded by a track. Each straightaway is 425 feet. The distance across the field (top to bottom in the diagram) is 150 feet. Find the area of the field.

a. Label the figure b. Write the appropriate formulas c. Substitute and solve d. Answer the question

a. Label the figure b. Write the appropriate formulas c. Substitute and solve d. Answer the question

straightaway

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THE GRADUATION CELEBRATION

You are planning a graduation celebration for 75 students in your class. The room for the party has dimensions of 60 feet by 90 feet. You will construct a stage in the shape of a trapezoid with parallel edges that are 10 feet and 20 feet. The depth of the stage will be 15 feet. You want to put circular tables with diameters of 10 feet around the room, and you need to leave space for chairs and for people to walk around. Each table can seat up to 9 people. You also want to have a big square dance floor. 1. Use the dot paper on the next page to make a scale drawing of the room, and lay out

where you will put the stage, dance floor, and tables. Label everything, and indicate how many people will sit at each table. (Hint: You may want to make a rough sketch of the room on dot or grid paper, and then transfer it onto this packet.)

2. What is the area of the stage? 3. How many tables do you need? What is the area of the space taken up by the tables? 4. What are the dimensions of your dance floor?

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SCALE DRAWING OF THE ROOM Use this page to make a rough sketch of the room.

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SCALE DRAWING OF THE ROOM Use this page to make the final draft of your scale drawing.

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Circles 2.3 Vocabulary, Skill Builders, and Review


FOCUS ON VOCABULARY (GEO2) Select the word from the word bank that best completes the sentence.

1. The __________ of a circle is the distance around it.

2. A line segment with both endpoints on the circle is called a __________.

3. Points on a circle are equidistant from its __________.

4. A line segment from the center of a circle to any point on the circle is called

a ___________.

5. A __________ is the set of all points in a plane that are a given distance (radius) from a

given point (center).

6. A chord that goes through the center of a circle is called a __________.

7. 227

and 3.14 are approximations for __________.

Word Bank circle radius diameter chord circumference center of a circle

pi

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SKILL BUILDER 1

Round to the nearest hundredths. 1. 53.429 ≈__________ 2. 0.6539 ≈__________ 3. 100.985 ≈_________

Round to the nearest thousandths. 4. 0.35735 ≈_________ 5. 510.7441≈________ 6. 1.0004 ≈_________

Write the following numbers in words: 7. 30.19_________________________________________________________ 8. 85.073________________________________________________________ Compute.

9. × 1 .11 2.0

10. ×5.01 2.2 11. ×0.9 6.2

12. ÷15.68 0.2

13. 0.3 45.6 14. 62.80.04

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SKILL BUILDER 2 Compute. Simplify when possible.

1. ×537

2. ×4 727 9

3. ÷24 38 5

4. ×5 2

12 9

5. ÷1 16 5

6. ÷4 8

19 18

7. ÷7 52

14 21

8. ×3 45 9

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SKILL BUILDER 3 1. Find the circumference of a circle whose radius is 14 inches.

a. Write the solution as an exact answer.

b. Write the solution as an approximation using π ≈ 3 .

c. Write the solution as an approximation using π ≈ 3.14 .

d. Write the solution as an approximation

using π ≈ 227

.

2. Find the radius, circumference, and area of a circular table whose diameter measures 3.4

meters. Write your solutions as exact answers. Radius: ____________ Circumference: _____________ Area: ______________ 3. Pizza House sells 16-inch-diameter pizzas and 12-inch-diameter pizzas. How much more

pizza would you get by ordering the larger pizza than the smaller one?

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This page is left intentionally blank for notes.

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TEST PREPARATION (GEO2)

Show your work on a separate sheet of paper and choose the best answer. 1. What is the circumference of a round table with a radius of 2.5 feet?

A. ≈ 5C ft B. ≈ 7.85C ft C. ≈ 15.7C ft D. ≈ 19.625C ft 2. What is the circumference of a circular rug with a radius of 1.5 yards?

A. 9.42 yards B. 7.07 yards C. 4.71 yards D. 3 yards 3. The circumference of a bicycle wheel is 50.24 inches. What is the diameter? Use 3.14

for π .

A. 32 inches B. 16 inches C. 8 inches D. 4 inches 4. If a circle has a radius of 7 meters, what is the area of half of the circle?

A. 76.9 m 2 B. 153.9 m 2 C. 44 m 2 D. 22 m 2 5. A pizza has a diameter of 18 inches. The pizza is cut into eight equal pieces. What is

the area of each piece? Round to the nearest tenth.

A. 254.3 sq. in B. 31.8 sq. in C. 56.5 sq. in D. 7 sq. in 6. What is the area of the circle rounded to the nearest tenth?

A. 8.5 cm B. 17 cm C. 22.9 cm D. 91.7 cm

5.4 cm

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This page is left intentionally blank for notes.

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KNOWLEDGE CHECK (GEO2)

Show your work on a separate sheet of paper and write your answers on this page. 2.1 Circumference 1. Find the circumference of a circle whose radius is 8 meters.

Approximate the measurement using π ≈ 3

Approximate the measurement using π ≈ 3.14

Approximate the

measurement using π ≈227

2. What is the circumference of a circle with a radius of 3 meters?

3. The circumference of a compact disc is 28.26 centimeters. What is the diameter? 2.2 Area of Circles 4. Find the area of a circle with a radius of 12 km.

5. The area of a circle is 50.24 sq. cm. What is the radius of the circle? 6. Explain which has a greater area, a circle with radius 3 meters or a square with side length

3 meters.

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Circles


Home-School Connection (GEO2)

Here are some questions to review with your young mathematician. 1. The diameter of a nickel is 2 centimeters. What is the circumference? 2. What is the circumference of a circle with a radius of 3.5 meters? 3. What is the circumference of a plate with a radius of 7 in? What is the area of the plate?

Use π ≈227

to find the approximation of the measurement.

4. Which has a greater area, five circles, each with the radius 1 meter, or one circle with radius 5 meters. Parent (or Guardian) signature ____________________________

Selected California Mathematics Content Standards

AF 6.3.1 Use variables in expressions describing geometric quantities (e.g. P = 2w + 2l, A = 12

bh, C = π d – the

formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).

MG 6.1.2 Know common estimates of π (3.14; 227

) and use these values to estimate and calculate the circumference

and the area of circles; compare with actual measurements. NS 7.1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and

applications. MG 7.2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface

area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

MG 7.2.2 Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects.

FIRST PRINTING DO NOT DUPLICATE © 2009

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Only Preview For · 2012-03-23 · Circles 2.1 Circumference Geometry and Measurement Unit (Student...

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