Chapter 10 Polygons and Circles - Mangham Mathmanghammath.com/Chapter Packets/Chapter 10 Polygons...

Created by Lance Mangham, 6th grade math, Carroll ISD

ACCELERATED MATHEMATICS

CHAPTER 10

MEASUREMENTS OF POLYGONS AND CIRCLES

TOPICS COVERED:

• Perimeter of polygons

• Area of rectangles and squares

• Area of parallelograms

• Area of triangles

• Area of trapezoids

• Circle vocabulary: radius, diameter, chord, tangent, secant

• Discovering pi

• Circumference of circles

• Area of circles

• Area of composite shapes


Accelerated Mathematics Formula Chart Name:

Linear Equations

Slope-intercept form y mx b= + Direct Variation y kx= (8th grade)

Constant of proportionality y

kx

= Slope of a line 2 1

2 1

y ym

x x

−=

− (8th grade)

Circumference Circle 2C rπ= or C dπ=

Area

Rectangle A bh= Trapezoid 1 2

1( )

2A b b h= +

Parallelogram A bh= Circle 2A rπ=

Triangle 2

bhA = or

1

2A bh=

Surface Area (8th grade) Lateral Total

Prism S Ph= 2S Ph B= +

Cylinder 2S rhπ= 22 2S rh rπ π= +

Volume

Triangular prism V Bh= Cylinder 2 or V Bh V r hπ= = (8th grade)

Rectangular prism V Bh= Cone 21 1 or

3 3V Bh V r hπ= = (8th)

Pyramid 1

3V Bh= Sphere 34

3V rπ= (8th grade)

Pi 22

3.14 or 7

π π≈ ≈

Distance d rt= Compound Interest (1 )tA P r= +

Simple Interest I prt= Pythagorean Theorem 2 2 2a b c+ = (8th grade)

Customary – Length 1 mile = 1760 yards

1 yard = 3 feet 1 foot = 12 inches

Customary – Volume/Capacity 1 pint = 2 cups

1 cup = 8 fluid ounces 1 quart = 2 pints

1 gallon = 4 quarts

Customary – Mass/Weight 1 ton = 2,000 pounds 1 pound = 16 ounces

Metric – Length 1 kilometer = 1000 meters 1 meter = 100 centimeters

1 centimeter = 10 millimeters

Metric – Volume/Capacity 1 liter = 1000 milliliters

Metric – Mass/Weight 1 kilogram = 1000 grams 1 gram = 1000 milligrams


AREA OF TRIANGLES and QUADRILATERALS

Area of a square Area of a rectangle Area of a parallelogram Area of a trapezoid

A l w= •

2A s

OR

A l w

=

= •

A b h= •

Area of triangle

1

2 2

bhA bh A= =

1 2

1 2

1( )

2

( )

2

A b b h

b b hA

= +

+

=

Area of trapezoid


Area/Volume/Surface Area Computation Page

EXAMPLES

1 2

2

1( )

2

1(10 20) 6

2

90 cm

A b b h

A

A

= +

= + •

=

2

2

3

3.14 10 5

1570 m

V r h

V

V

π=

= • •

=

2

2

2(8 6) (28) 10

376 in

S B Ph

S

S

= +

= • + •

=


Activity 10-1: Perimeter Name:

Perimeter: The distance around the outside of a figure. Per means around. Meter means measure. Thus, the perimeter of a figure is the measure around it. Find the perimeter of each figure. 1. 2. 3.

Find the missing part of each rectangle.

4. P = 160 mm

l= 32 mm w = 5.

P = 21.8 km w = 4.7 km

l =

Find the perimeter of each regular polygon with the side length listed.

6. regular hexagon / 28.5 millimeters

7. regular 12-gon / 3.25 yards

8. regular heptagon /

10.75 feet

9. regular 25-gon / 6 inches

Two rectangles are shown below. The value of x is the same for both rectangles.

10. What is the perimeter of rectangle I? A 10 x+ B 10 2x+ C 20 2x+

11. Write an expression that represents the perimeter of rectangle II.

12. If x is an integer, what are the smallest and largest values x can be?

13.

Suppose the areas of the two rectangles are equal. What is the value of x? What is the perimeter of each rectangle? What is the area of each rectangle?

2.5 km

2.4 km

2.0 km 2.9 km

1.6 km

Regular polygon

24 in

0.9 m

1.8 m

1.8 m

I

4

II

7


Activity 10-2: Area of Rectangles Name:

Show all work on separate paper including three steps for each problem: write the correct

formula, fill in the numbers for the variables, and then solve the equation. Game Shape Dimensions Perimeter Area

1. Racquetball Rectangle w = ft. l = 40 ft.

A = 800 sq. ft.

2. NCAA basketball Rectangle w = 50 ft. l = 94 ft.

3. Ice hockey Rectangle w = 85 ft. l = ft.

P = 570 ft.

4. Volleyball Rectangle w = ft. l = 60 ft.

A = 1800 sq. ft.

The Exxon Valdez oil spill in Prince William Sound, Alaska, in 1989, covered 7 million acres. How large is an acre? Use a calculator for these questions. If necessary round to the nearest tenth.

5. A football field is 300 feet long and 160 feet wide. An acre is 43,560 square feet. Is a football field smaller or large than an acre?

6. How many acres are in a football field?

7. Imagine a square with an area of 20 acres. What is the length of a side of a square, in feet, that has an area of 20 acres?

8. The length of a football field is 100 yards. How many football field lengths is the side of a 20 acre square?

9. The Exxon Valdez oil spill covered more than 7 million acres. What is the length of a side, in feet, that has an area of 7 million acres?

10. How many football field lengths is the side of a 7 million acre square?

11. There are 5280 feet in a mile. How many miles in length is the side of a 7 million acres square?

A NBA backboard is show at the right.

12. What is the area of the entire backboard?

13. What is the area of the inner rectangle on the backboard?

14. What is probability that a random shot that hits the backboard will hit the inner rectangle?


Activity 10-3: Area of Parallelograms Name:

Formula for the area of a parallelogram: A bh= The height is measured straight up from the base. The height of this parallelogram is 4 m.

2

8 4

32 m.

A bh

A

A

=

= •

=

Find the perimeter and the area of each parallelogram. For the area, show all steps. 1. 2. 3. 4. 5. 6.

7. The base of a parallelogram is 10 in. The height is 2 in. more than half the base. Find the area.

8. The height of a parallelogram is 4.5 cm. The base is twice the height. What is the area?

9. The area of a parallelogram is 60 ft.2 The height is 5 ft. How long is the base?

10. The area of a parallelogram is 275 cm.2 The base is 25 cm. Find the height.

11. Mr. Mangham wants to figure out how many bags of fertilizer he needs to cover his yard. You know the following: the area of the yard, area each bag of fertilizer can cover, cost of each bag, weight of each bag. How would you determine the number of bags needed?

5 m

8 m

4 m

16 ft

8 ft 10 ft m

m

15 m

3.6 cm

5.1 cm

3.2 cm

9.3 cm

7.5 cm 6.5 cm

100 ft

90 ft 90 ft 2.0 m 1.8 m

0.7 m


Starting tonight we are moving to formulas that require more than one step. For shapes with more than one step, we have

an organized way to show all our work.

It is 3 easy steps:

• write the formula

• fill in the numbers you know

• solve

Example:

2

2

6 5

2

15 in.

bhA

A

A

=

•=

=

Be ready to turn this work on a separate sheet of paper for

all future homework assignments. This is the process we will use no matter how easy or difficult the numbers are. --------------------------------------------------------------------------- I understand to receive credit for all area (except for rectangles and parallelograms), volume, and surface area problems that I must show the three steps listed above on a separate sheet of notebook paper.

Name/Signature: ____________________________________

6 in

5 in


Activity 10-4: Area of Triangles Name:

Formula for the area of a triangle: 1

or 2 2

bhA bh= (Half of the formula for a parallelogram.)

The height is measured straight up from the base. The height of this triangle is 5 in.

2

2

6 5

2

15 in.

bhA

A

A

=

•=

=

Find the area of each triangle using the formula above. Show all steps on a separate sheet of paper. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12 yd. 23.7 km.

15 yd.

19 km

11.2 km.

16.9 km

8 mm

5 mm

in

in

12 yd

12 yd

6 in

2.5 in

4.5 km

1.4 km

6 in

5 in

72 in.

18 cm

7.5 cm

m

m

58 in.


Activity 10-5: Area of Trapezoids Name:

Formula for the area of a trapezoid: 1 2

1( )

2A b b h= + [ 1 2

1( )

2b b+ is just the average of the two bases.]

Example: The two bases are always parallel to each other.

Find the area of each trapezoid using the formula above. Show all steps on a separate sheet of paper. Trapezoid A Trapezoid B Trapezoid A

1. x=14 cm, y=26.5 cm, z=12 cm 2. x=4 cm, y=10 cm, z=5 cm

3. x=40 m, y=50 m, z=20 m 4. x=7 ft, y=15 ft, z=7 ft

Trapezoid B

5. x=6 in, y=16 in, z=9 in 6. x=41 cm, y=78 cm, z=22 cm

7. x=2.8 m, y=2.5 m, z=1.5 m 8. 1 3

=2 in, =12 in, =9 in4 4

x y z

9.

Cassie draws the following 4 figures. List the shapes in order of area from greatest to least.

10. What happens to the area of a trapezoid if both bases are tripled?

11. What happens to the area of a trapezoid if both bases and the height are all divided by 3?

12 in

15 in

6 in

x

y

z

x

y

z

6 cm

10 cm

10 cm

12.5 cm

5 cm

8 cm

10 cm

8 cm

1 2

2

1( )

2

1(12 15)6

2

1(27) 6

2

81 in.

A b b h

A

A

A

= +

= +

= • •

=


Activity 10-6: Area of Composite Shapes Name:

Find the area of each figure. Show all steps. 1. 2. 3. 4. 5. Find the area of the shaded region in each figure. 6. yard with a sandbox 7. wall with windows 8. sidewalk around pool Yard: 15 ft by 20 ft Wall: 8 ft by 16 ft Sandbox: 6 ft by 7 ft Each window: 5 ft by 4 ft Sidewalk: 30 ft by 30 ft Pool: 27 ft by 27 ft

9. A bedroom is 15 ft long and 12 ft wide. How much will it cost to carpet the room if carpeting costs $22 per square yard? (1 yd = 3 ft)

10. A rose garden in the city park is rectangular and is 9 m wide. If the area of the rectangle is 144 m2, what is the length of the garden?

11. Cindy had a rectangular garden last year with an area of 60 sq. ft. This year the garden is one foot wider and three feet shorter than last year, but it has the same area. What were the dimensions of the garden last year?

12.

An average gallon of paint will cover 350 sq. feet of wall or ceiling space. All of your ceilings are 8 feet high. Your living room is 22 feet by 18 feet. Your kitchen is 15 feet by 15 feet and your dining room is 12 feet by 14 feet. How many gallons of paint would you need to give one coat of paint to each wall and ceiling? If paint costs $23 a gallon, what would the total cost be?

6 m

4 m

2 m

2 m 4 m

6 m

3 m 2 m

6 in 8 in

10 in

5 in

12 in

12 in

7 in

7 in

9 cm

10 cm

5 cm 18 cm


Activity 10-7: All Area Name:

1.

Ms. Wagner painted the outside of the patio door to her house, as shown below. She did not paint the window or the doorknob. Which is the closest to the painted area of the door in square feet?

A. 231 ft. B.

228 ft. C. 225 ft. D.

218 ft.

2.

A pest-controlled company was hired to spray the lawn represented by the shaded region shown below. What was the area in square feet that was sprayed?

A. 219,280 ft. B.

220,000 ft. C. 237,680 ft. D.

217,680 ft.

3. A triangular sail has a base of 5 m and a height of 10 m. If canvas costs $18 a square meter, find the cost of canvas to make the sail.

4. A square dinner napkin 8 in. on each side is folded along its diagonal. Find the area of the folded napkin.

5. A shuffleboard court is a large isosceles triangle with b = 6 ft and A = 27 ft2. What is the length (height) of the shuffleboard court?

6. If you doubled the height of a triangle, what would happen to the area of the triangle?

7. If you doubled both the base and the height of a triangle, what would happen to the area?

8.

The area of ABC∆ is greater than the area of DEF∆ . Which must be true?

A. The height of ABC∆ is greater than the height of DEF∆ .

B. The perimeter of ABC∆ is greater than the perimeter of DEF∆ .

C. The sum of the angles of ABC∆ is greater then the sum of the angles of DEF∆ .

D. Base and height: At least one of them is greater on ABC∆ .

4 ft.

7 ft. 2 in by 3 in

1 ft by 1 ft (each square)

200 feet

House Gar

100 feet

40 ft by 40 ft

24 ft by 30 ft


Activity 10-8: Area and Perimeter Name:

Use graph paper for all drawings and all work.

1. Draw a figure whose perimeter is 24 units. 2. Draw a different figure whose perimeter is also 24 units. 3. Draw a figure whose area is 24 square units. 4. Draw a different figure whose area is also 24 square units. 5. Can two different figures have the same area but different perimeters? Explain your answer. 6. Your dog, Benji, needs a new play area. You are in charge of building a fence around the dog’s

play area so that he can’t run away. You are given 80 feet of fencing to build your play area. Build two different play areas, each using 80 feet of fencing, which you think would be suitable for a dog using all of the fencing. For each of your SCALE drawings:

• Calculate the perimeter

• Calculate the area

• Explain why/how the shape you chose would be good for a dog’s play area

PART 2

7. The perimeter of the rectangle is 62 in. Find the length of each side.

8. What is the area of a rectangle that is formed by connecting the points (1,2), (5,2), (5,4), and (1,4) on a coordinate grid?

9. What is the area of a triangle with vertices (1,2), (7,2), and (3,6)?

10. Amanda bought 40 meters of fencing to make an enclosure for her dog, Sushi. If Amanda expects a rectangular enclosure, what is the largest area it can have? Explain your answer.

11. The width of a rectangle is 4.5 inches and its perimeter is 31 inches. What is the length of the rectangle?

PART 3

12. The club house is a rectangle that is 25 feet by 40 feet in size. The officers voted to put a 6-foot sidewalk all around the building, leaving a 2-foot space for plants between the building and the sidewalk. Give the perimeter of the outer edge of the sidewalk and the area of the sidewalk itself. 13. What is the area of each black and white piece if the whole square measures 20 cm on each side? What percent of the area of the large square is the small shaded square?

17 in.


Activity 10-9: Circles Name:

Diameter: a segment that passes through the center of the circle and has both endpoints on the circle Radius: a segment that has one endpoint at the center of the circle and the other endpoint on the circle Center: the middle point of a circle Semicircle: half of a circle Arc: part of a circle

The diameter is twice the length of the radius.

The radius is half the diameter.

1. Draw and label the following parts (one per circle).

Diameter KL

Center O

Radius OT Central angle WOK∠

Find the unknown length of each circle.

Radius Diameter Radius Diameter

2. 8 cm. 3. 110 in.

4. 48 ft. 5. 23 mm.

6. 55 ft. 7. 18 in.

8. 5

68

yd. 9. 3

74

in.

10. 12.3 mi. 11. 16.4 m.

12. 3.3 cm. 13. 1.25 yd.

14. 2.6 m. 15. 9.1 in.

Symbols

Angle BAC BAC∠ Ray AB AB��

Line Segment AB AB Arc AB �AB

Line AB AB��

B

P

A O

Point O is the center of the circle.

AB is the diameter.

OA is the radius. OP is also a

radius.

A

P

O

Q

and are central angles.AOP AOQ∠ ∠

2

dr =2d r=


Activity 10-10: Circles Name:

How can I remember the difference between diameter and radius?

Radius comes from the same root word as “ray”. A ray of sunshine starts at the center of the sun and goes out from that point. The radius of a circle starts at the center of the circle!

Solve the following problems using the circle above.

1. The points on a circle are all the same distance from the...

2. A line segment from the center to any point on the circle is a…

3. A diameter of the circle in the drawing above is the segment…

4. Which of the following is not a radius: , , or ?OA OD BC

5. Part of a circle, such as between points B and C, is an…

6. An angle whose vertex is at the center of a circle is a…

7. Which of the following is not a central angle:

, , or ?AOD COD BCA∠ ∠ ∠

8. Points A, B, C, and D are all the same __________ from point O.

9. If the length of AC is 20 cm, then the length of OC is…

10. If the length of OA is 20 cm, then the length of OD is…

11. If the length of OD is 20 cm, then the length of AC is…

12. The length of a radius is __________ the length of a diameter.

13. The set of points in a plane at a fixed distance from a given point is a…

B

C

O

A

D


Activity 10-11: Measuring Circles Name:

Measure at least 6 of the objects provided by your teacher. For each object record the information below. You may use a calculator as needed.

Item Name

Diameter

(measure in mm)

Circumference

(measure in mm)

Radius

(calculate in mm)

Diameter

nceCircumfere

(round to the nearest ten-thousandth)

***** MEAN *****

***** MEDIAN *****

***** RANGE *****

***** CLASS MEAN *****

***** CLASS MEDIAN *****


Activity 10-12: Extraordinary Pi Name:

• Pi is the number of times a circle’s diameter will fit around its circumference.

• Here is pi to 64 places: 3.1415926535897932384626433832795028841971693993751058209749445923

• Pi occurs in hundreds of equations in many sciences including describing DNA, a rainbow, ripples where a raindrop fell into water, distribution of prime numbers, geometry problems, waves, navigation, etc.

• Half the circumference of a circle with radius 1 is exactly Pi. The area inside that circle is also exactly Pi!

• Taking the first 6,000,000,000 decimal places of Pi, this is the distribution: 0 occurs 599,963,005 times 1 occurs 600,033,260 times 2 occurs 599,999,169 times 3 occurs 600,000,243 times 4 occurs 599,957,439 times 5 occurs 600,017,176 times 6 occurs 600,016,588 times 7 occurs 600,009,044 times 8 occurs 599,987,038 times 9 occurs 600,017,038 times

• Pi is irrational. An irrational number is a number that cannot be expressed in the form (a / b) where a and b are integers.

• The Babylonians found the first known value for Pi in around 2000BC; they used (25/8) The Egyptians used Pi = 3 but improved this to (22 / 7). They also used (256/81).

• In around 200 BC Archimedes found that Pi was between (223 / 71) and (22 / 7). His error was no more than 0.008227%. He did this by approximating a circle as a 96 sided polygon.

• Ludolph Van Ceulen (1540 - 1610) spent most of his life working out Pi to 35 decimal places. Pi is sometimes known as Ludolph's Constant.

• The first person to use the Greek letter was Welshman William Jones in 1706. He used it as an abbreviation for the 'periphery' of a circle with unit diameter.

• The Pi memory champion is Hiroyoki Gotu (21 years old) who memorized an amazing 42,000 digits.

• Pi was calculated to 2,260,321,363 decimal places in 1991 by the Chudnovsky brothers in New York.

• Most people would say that a circle has no corners - but it is more accurate to say that it has an infinite number of corners.

• At position 762 there are six nines in a row. This is known as the Feynman Point.


CIRCLES

Important information about circles:

• The diameter is exactly twice the radius. The radius is exactly half of the diameter.

• π is a number that goes on forever. π represents how many diameters it takes to go around the

circumference of any circle. π is an irrational numbers. It is approximately 3.14 as a decimal

and it is approximately 22

7 as a fraction.

Important formulas for circles:

Radius to diameter: 2

dr = Diameter to radius: 2d r=

Circumference: 2C rπ= or C dπ= Area: 2A rπ=

RADIUS

DIAMETER

I know the RADIUS and

I want to know the DIAMETER

***** MULTIPLY BY 2 *****

If radius = 5, then the diameter = 10.

No steps needed.

I know the DIAMETER and

I want to know the RADIUS

***** DIVIDE BY 2 *****

If diameter = 14, then the radius = 7.

No steps needed.


I want to know the CIRCUMFERENCE

If the radius of this circle is 9,

2

2 3.14 9

56.52

C r

C

C

π=

= • •

=

I know the DIAMETER and I want to know

the CIRCUMFERENCE

If the diameter of this circle is 6,

3.14 6

18.84

C d

C

C

π=

= •

=


I want to know the AREA

If the radius of this circle is 4, 2

23.14 4

50.24

A r

A

A

π=

= •

=

I know the DIAMETER and

I want to know the AREA

If the diameter of this circle is 16, then the radius is 8.

2

23.14 8

200.96

A r

A

A

π=

= •

=


Activity 10-13: Circumference and Area Name:

Circum means around or bend around. Ferre means to bring or carry. The circumference of a circle is the measure you need to get around the entire circle. Circumnavigate means to sail around the globe.

223.14

7π ≈ ≈

Important formulas for circles:

Radius to diameter: 2

dr = Diameter to radius: 2d r=

Circumference: 2C rπ= or C dπ= Area: 2A rπ=

Find the circumference and area of each circle given the diameter or radius. Use 3.14 or 22

7for π .

Show all work on separate paper including three steps for each problem: write the correct

formula, fill in the numbers for the variables, and then solve the equation.

Radius Circumference Area Diameter Circumference Area

1. 3 in 2. 8 yd

3. 211

yd14

4. 1.2 ft

5. 37.68 mi 6. 24 yd

7. 55 m 8. 400π ft2

9. 3

4 cm 10. 20π cm

11. 7.1 cm 12. 157 in

13. 1

22

cm 14. 16 in

15. 60π m 16. 32 in

17. x cm 18. y in

19-20.

Write an expression to find the circumference of the outside of the donut. The radius of the inner circle is 2 cm. Do not solve.

Write an expression to find the area of the donut. The radius of the inner circle is 2 cm. Do not solve.

21. If a 14 inch pizza has a 1 inch crust, what is the area of the crust? To the nearest percent, what percent of the pizza is crust?

9 cm


Activity 10-14: Circumference and Area Name:

Coin Diameter Weight

Penny 19 mm 2.5 g

Nickel 21 mm 5 g

Dime 18 mm 2.3 g

Quarter 24 mm 5.7 g

1. Determine the circumference of a penny.

2. Determine the area of a penny.

3. A row of pennies set end to end is 1.52 meters long. How many pennies are there?

4. A group of pennies weighs 2,500 kilograms. How many pennies are there?

5. Determine the circumference of a nickel.

6. Determine the area of a nickel.

7. A row of nickels set end to end is 35.7 centimeters long. How many nickels are there?

8. A group of pennies is worth $4.75. How much do they weigh?

9. Determine the circumference of a dime.

10. Determine the area of a dime.

11. A group of dimes weighs 172.5 grams. How many dimes are there?

12. A stack of dimes weighs 345 grams. How much money is there in the stack?

13. How much more money is 138 grams of dimes than 275 grams of nickels?

14. Determine the circumference of a quarter.

15. Determine the area of a quarter.


Activity 10-15: Inuit Architecture Name:

The Inuit are Native Americans who live primarily in the arctic regions of Alaska and Canada. The Inuit word iglu means “winter house”. Later the term came to mean a domed structure built of snow blocks, as shown in the figure at the right. Several families would build a cluster of iglus that were connected by passageways and shared storage and recreation chambers. Use the drawing to answer the following questions. Use 3.14 for pi.

1.

List the radius of each of the chambers.

2.

Determine the circumference of each chamber.

3.

Determine the area of each chamber. Then calculate the total area of all chambers.

4.

Estimate the distance for the front of the entry chamber to the back of the storage chamber.

5.

If Mr. Mangham runs around the storage iglu 5 times, how far will he run?

Storage d = 10

ft.

d = 9

ft.

d = 9

ft.

d = 8 ft.

d = 8 ft.

Recreation d = 12 ft.

Entry d = 6 ft.

6. A person wishes to place a covering on the floor in the entry, recreation, and storage rooms. If coverings are sold in 40 square feet pieces, how many pieces will she need? 7. If the radius of one of the iglus is tripled, then the area of the iglu is

multiplied by….


Activity 10-16: Bull’s Eye Name:

1. The radius of the inner circle is one inch. Each successive circle has a radius increase of one inch, so the outside circle has a radius of five inches (note that they are not drawn in actual size).

Which has a greater area – the shaded region or the striped region? 2. Which shape has more area: the one on the left or the one on the right? 3. The Square Pizza Company only serves pizza in the shape of squares. Unfortunately, an employee mistakenly ordered 16-inch diameter circular pizza pans. What is the largest size square pizza that can be made from this pan? Hint: Make triangles 4. The Square Pizza Company makes pizza dough to place in 16-inch diameter circular pans, then cuts off the edges of the circle to create the largest square pizza possible. If the area of the cut-off dough can be reformed into another square pizza, then what side length would this recycled square pizza have? You can list your answer as between two whole numbers.


Activity 10-17: The Margaret Hunt Hill Bridge Name:

The Margaret Hunt Bridge was completed near downtown Dallas in March 2012, almost five years after construction began. The bridge cost $182 million to build and it was made out of 11,643,674 pounds of structural steel. It is a cable stayed bridge, not a suspension bridge.

The bridge cables are two different sizes. There are 44 large cables that have a diameter of 6.3 inches and are made up of 31 individual strands of wire. The 14 smaller cables have a diameter of 4.9 inches and are made up of 12 individual strands of wire. You may use a calculator for all computation, but show all formula steps as usual.

1. What is the circumference of each of the large cables?

2. What is the area of a cross-section of one of the large cables?

3. What is the circumference of each of the smaller cables?

4. What is the area of a cross-section of one of the smaller cables?

5. Assuming the strands of wire inside the cables take up 90% of the area, what is your best guess at the diameter and circumference of a single strand of wire inside the larger cable?

6. How does the circumference of a circle change if you double the length of its radius?

7. How does the circumference of a circle change if you double the length of its diameter?

8. If you triple the length of the radius of a circle, how does its circumference change?

9. If you make the circumference of a circle half as long, how does its diameter change?

10. If you multiply the radius of a circle by y, how does its diameter change?

11. A circular swimming pool has a border of 30 curved tiles. There are blue and white tiles in a pattern of two blue tiles between every white tile. How many white tiles are there?


Activity 10-18: Do You Want to Build a Snowman? Name:

You may use a calculator for this activity. Do not use the pi button – use 3.14 for pi.

For all problems involving an equation (circumference, area, etc.) show all steps.

For all problems asking “how many turns” your answer should be a whole number.

Fresh snow has just fallen and Ashley decides to make a snowman. Ashley makes a snowball 6 inches in diameter. She then rolls the snowball one full turn.

1. Draw a picture of the original

snowball and draw and label the diameter.

2.

How far does Ashley roll the snowball? Show all equations and all steps in the box at

the right.

3.

Ashley finds that 3 inches of new snow sticks as she rolls the

snowball (3 inches added to every part on the original

circle). Draw a picture of the new snowball.

4.

What is the new diameter of the snowball after Ashley rolls it one full turn?

(Hint: It is not 9 inches.)

5.

How many total turns will Ashley have to make so that the diameter of the snowball becomes 72 inches?

(Hint: The first 6 inches did not require any turns.)

6.

How many total turns will it take for the circumference to become greater than 300 inches?

(Hint: Start with the circumference formula to find d.)

Show your original equation and all steps in

the box at the right.


7.

How many total turns will it take for the area to become greater than 1000 in.2?

(Hint: Start with the area formula to find r.) Show all equations and all steps in the box at

the right.

8.

Ashley makes a second snowball and then rolls it one full turn. Again 3 inches of snow stick to it. She measures the new circumference (C) to be about 44 inches. To the nearest inch, what is the diameter of Ashley’s rolled snowball? Show all equations and all steps in the box at

the right.

9.

Using the diameter from #8, what is the area of the rolled snowball? Show all equations and all steps in the box at

the right.

10. What was the diameter of the snowball before Ashley rolled it?

11.

What is the area of the snow that stuck to the original snowball in #8? Show all equations and all steps in the box at

the right.

12.

Ashley rolls a huge snowball 4 feet in diameter. She decides to make another

snowball 2

3 of its diameter to put on top. In inches, what is the diameter of

the snowball Ashley wants to put on the top?

13.

In square inches, what is the area of both snowballs combined? Show all equations and all steps in the box at

the right.

14. Ashley started with a snowball 8 inches in diameter. How many complete turns will Ashley need to roll it before she has a snowball the size she wants to put on top?


Activity 10-19: More Circles Name:

1. Draw a circle similar to the one below with a dot in the middle without lifting your pencil from the paper. Can it be done? 2. A pizza parlor has 3 size pizzas. Would you get more if you had one small pizza, three-fifths of a medium pizza, or one-third of a large pizza?

SIZE DIAMETER

Small 10 inch diameter

Medium 14 inch diameter

Large 18 inch diameter

3. In the picture below, the circle’s area is what percent of the larger squares area? If the larger square has an area of 10 square inches, what is the area of the smaller square? 4. Queen Dido’s Cutting Trick: Take a 3x5 index card. Fold the card in half hamburger style. Start your first cut on the fold just barely in from the side. Cut almost all the way up to the top. Next cut from the top almost all the way back to the bottom. Continue until you end up with an odd number of cuts. Unfold the card and cut all the folds except for the ones at the end of the card. You will have a circle big enough to step through! 5. Draw two squares so that each dot is completely contained in a region by itself.


Activity 10-20: Hot Tubs Name:

Hot tubs and in-ground swimming pools are sometimes surrounded by borders of tiles. You have a square hot tub with sides of length b feet. Your tub is surrounded by a border of square tiles. Each border tile measures 1 foot on each side. How many 1-foot square tiles will be needed for the square border of the hot tub that has edge length of b feet? Draw separate pictures in which you group the border tiles in different, logical arrangements and then express the total number of tiles needed with equivalent expressions (one for each picture you have drawn).

This problem tells you specifically to draw pictures, so that must be the strategy! Luckily for you, the pictures have already been drawn for you on the other side. Your job is to look specifically at how the tiles have been grouped to come up with an appropriate expression using numbers, variables, and operations. Hint: If you simplified all your expressions, you would get the same answer every time.

THE HOT TUB

b

b

1

1


Four segments + Four corners Two long segments + Two short segments

Four segments – Four corners All shading – Inside shading (since they are covered twice)

Can you think of any more logical configurations? If so, draw them and determine their formulas. Four segments

Chapter 10 Polygons and Circles - Mangham Mathmanghammath.com/Chapter Packets/Chapter 10 Polygons...

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Transcript of Chapter 10 Polygons and Circles - Mangham Mathmanghammath.com/Chapter Packets/Chapter 10 Polygons...