Thrill U. THE PHYSICS AND MATH OF AMUSEMENT … education... · THE PHYSICS AND MATH OF AMUSEMENT...

Thrill U. THE PHYSICS AND MATH OF AMUSEMENT PARK RIDES

Geometry

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

© Copyrighted by Dr. Joseph S. Elias. This material

is based upon work supported by the National

Science Foundation under Grant No. 9986753.

Dorney Park/Kutztown University

Thrill U. - Geometry

Introduction

Welcome to Thrill U.!

This set of mathematics activities focuses on Geometry. We believe there is something for everyone as the collection represents a breadth of adventures. Collectively, the geometry activities span a wide range of topics found in informal geometry, plane geometry, analytic geometry and foundations of trigonometry. Some activities require straightforward data collection and calculations, many require collaboration and some involve interdisciplinary considerations. Several of the activities can be easily adapted for special needs students, some can be adapted for use as follow-up activities, and still others are appropriate for schools with intensive scheduling programs.

Each activity is preceded by an "Information Sheet" to help guide teachers in selecting appropriate activities for their particular group(s) of students. In addition to identifying objectives based on state and national mathematics standards, these pages provide a list of equipment needs and suggestions that may lead to the activity's successful completion. Teachers should feel welcome to adapt activities to the specific needs of their students. Teachers may request a “solution manual.” Contact Dr. Joseph S. Elias at [email protected].

Join us in May and challenge your students to experience geometry in action! Dr. Kathleen Dolgos Dr. Deborah Frantz Professor Emeritus Professor Emeritus College of Education College of Liberal Arts and Sciences Kutztown University Kutztown University

mailto:[email protected]

Thrill U.

Table of Contents

Acknowledgments Page i

Tips for Mathematics Teachers Page ii

Things to Bring/Dorney Park Information Page iii

Algebra Activities (separate manual) Pages 1 - 60

Geometry Activities

Specific Rides:

The Antique Carrousel Page 61

The Ferris Wheel Page 69

The Sea Dragon Page 75

Steel Force Page 81

White Water Landing – The Bridge Page 85

Features in the Park:

A Geometric Walk Page 91

Polygons in the Park Page 98

Rides That are Inscribed Page 103

The Fish Pond Page 110

i KUTZTOWN UNIVERSITY OF PENNSYLVANIA

Dorney Park/Kutztown University

Acknowledgments

During the winter of 1997, area teachers of physics and mathematics, professionals

from Dorney Park, and faculty from Kutztown University gave birth to Thrill U.: The

Physics and Math of Thrill Rides. In May 1999, the Thrill U. - Physics Planning

Committee presented physics activities in the form of Thrill U.

The Thrill U. - Mathematics Planning Committee was established in the fall of

1999. The Committee began by creating algebra activities that would help teachers achieve

goals set forth by the state and national “Mathematics Standards.” As a result, the

mathematics component of Thrill U. consisted of algebra activities that were introduced in

May 2001. Geometry activities have been developed and have been a part of the

mathematics component since May 2002.

Thrill U. is the culmination of effort and time of many people. Its existence would

not have been possible without the collaborative efforts of: the professional staff at Dorney

Park and Wildwater Kingdom; the administrators and academic faculty at Kutztown

University; teachers who had taken students to the Park and provided feedback; and (most

importantly) members of the planning committees. Each planning committee consists

primarily of area high school teachers of physics or mathematics. The leadership and

creativity of these teachers resulted in impressive sets of activities. Members of all

planning committees have worn out shoes in the Park, endured days of inclement weather,

fretted over success (or failure) of their students to complete preliminary versions of

activities while in the "piloting" stages, spent many hours in meetings, and countless hours

designing and editing the activities. In short, we admire and appreciate the efforts of all

who have contributed to the success of the Thrill U. project.

Geometry Planning Committee

Mrs. Susan Barnett Northwestern Lehigh High School

Ms. Karen Comegys William Allen High School

Mrs. Terri Costenbader Allentown Central Catholic High School

Dr. Kathleen Dolgos Kutztown University

Dr. Joseph Elias Kutztown University

Dr. Deborah Frantz Kutztown University

Mrs. Rose Gadbois Allentown Central Catholic High School

Mr. Keith Koepke Dorney Park and Wildwater Kingdom

Mr. Fran McGouldrick Allentown Central Catholic High School

Mrs. Brenda Snyder Kutztown University

Mr. Charles Waitkus William Allen High School

Tips for Mathematics Teachers

ii KUTZTOWN UNIVERSITY OF PENNSYLVANIA

Think of this as an adventure! To help make a "stress-free" day at the Park, we have created this list of

suggestions to guide you through your planning stage of Thrill U..

• Above and beyond all else, bring your sense of humor. Experienced teachers know that there will

be mistakes. Allow students to have fun as well as complete your selection of activities.

• While mathematics is an "exact" discipline, applications of mathematics are much "less exact."

ALL measurements and collected data will have inherent errors.

Accept it.

• Please do not forget copies of activity sheets, equipment and supplies. You might also consider

bringing a camcorder to record aspects of the rides for use in the classroom after Thrill U., or to

use as introductory preparation for next year.

• If your comfort level is low with orchestrating lab-type activities, consider consulting a science

teacher for assistance with logistics.

• Carefully peruse the complete list of activities and select those that will best fit the needs and

abilities of your students. (That is, do not expect your students to complete all of these activities!)

The difficulty levels are quite varied among the activities. Consider doing parts (but not all) of

some activities. You may modify them, or assign groups of students to them.

• Some activities take longer than others to complete. Keep in mind that it may be necessary to

observe, ride, or take measurements several times in order to obtain good data.

• As much as is feasible, introduce the students to the concepts to be studied during the weeks

leading up to the event. Consider planning time in class for calculations and analysis during the

days following the experience.

• In our opinion, students who may be fearful of some rides should not be forced to ride.

• Kutztown University students will serve as general assistants for you. They will be stationed at

designated Thrill U. rides from approximately 10:00 A.M. to 12:00 P.M. Inform your students

that they may ask the university students questions related to the activities. University students

will help students discover the "answers," but will not give them answers. Instruct your students

NOT to ask Dorney Park employees to give answers.

• Teachers are welcome to utilize a designated grove at Dorney Park to chat with other teachers

and members of the planning committees, or to use as a place for your students to work. Please

do not leave equipment and other valuables unattended at the grove.

Things to Bring

We present this list for your convenience and hope that it helps make your day at the Park

enjoyable as well as productive. You may wish to bring some or all of these items with

you to Thrill U.

iii KUTZTOWN UNIVERSITY OF PENNSYLVANIA

• tickets for you, your students and your chaperones

• copies of your selection of activities, enough for your group

• pencils and paper

• stopwatches

• calculators (depends on activities: basic, basic with tangent key, or graphing)

• camcorder

• clipboards

• inclinometers or protractors with a plumb bob

• appropriate clothing and perhaps a change of clothing

• sunscreen, hats, raincoats

• money for food, drinks, phone

• measuring tape or string

• masking tape

• backpacks or plastic bags to keep papers and equipment dry and together

• maps of the Park (can be picked up at the entrance to the Park)

• a good reserve of energy and enthusiasm for exploration

Dorney Park Information

General Information: (800) 551-5656 or (610) 395-3724

Group Sales Information: (610) 395-2000

For specific questions about ticket sales for Thrill U., call Matt Stoltzfus at (610) 392-

7607 or e-mail him at [email protected]

Visit our website: Dorney Park Thrill U.

https://www.dorneypark.com/youth-programs/youth-education

The Antique Carrousel

61 KUTZTOWN UNIVERSITY OF PENNSYLVANIA

Information Sheet

topics: central angle

interior and exterior angles

inscribed polygon

sector of a circle

arc length

objectives: to use the number of sectors in various circles to determine the

measure of central angles and number of sides of inscribed

polygons.

to examine the relationship between the number of sides of the

inscribed polygons, the measure of each central angle, and the

measure of each exterior angle of these polygons.

to approximate the arc length of a sector of a circle and use this to

calculate the radius of the circle using the formula

arc length = (measure of central ÷360°) 2r

equipment: pencil

calculator (basic four function, non-graphing type)

activity sheets

extra paper

25-foot tape measure

notes to the teacher:

Prior to your visit to Dorney Park, you may want to introduce the

Polygon Grid Activity. Introduce the vocabulary terms associated

with polygons. Please consider that all polygons in this activity are

regular polygons.

In addition to the Antique Carrousel, this activity includes the

circular rides Waveswinger, Enterprise and Apollo 2000.



Please notice that there are two carousels at Dorney Park. The Antique Carrousel

has 4 rows of horses mounted to the rotating platform. Be sure you are doing this

activity at the Antique Carrousel.

Before you get started on this activity, let’s define some terms.

A

B

C

D

Sector of a circle

CAB is a central angle.

CBD is an exterior angle of the polygon

inscribed in circle A. It is adjacent and

supplementary to interior CBE.

,AF AE , and arc FE bound sector FAE. A

slice of pizza is a good illustration of a sector

of a circle.

F

E



Study questions for the Antique Carrousel:

1. Observe the structure that encloses the Antique Carrousel and decide how many

sides are in this regular polygon. Calculate the measure of one central angle, the

measure of one interior angle and the measure of one exterior angle for this

polygon.

Number of

sides

in polygon

Measure of one

central

Measure of one

interior

Measure of one

exterior

2. Now look at the Antique Carrousel. As the ride turns, count the number of sectors

in the top of the carousel. Each sector is bounded by rows of lights.

How many sectors are there? _________________

Complete the grid:

Number of

sides

in polygon

Measure of one

central

Measure of one

interior

Measure of one

exterior

This is one sector

of the ride.



3. Now take a look at the fence around the Antique Carrousel.

Measure the distance between two adjacent fence posts. ___________________

We will use this distance as an approximation of the length of the arc for the circular area

bounded by the fence. We will assume that all fence sections are congruent and are the

same width as each of the 8 pillars, which hold up the roof over the carousel. There are

40 fence sections and 8 pillars.

Now that you know the arclength of 1/48th of this circle, we are going to find the length

of the radius of the circle.

a. First find the measure in degrees of the central angle for this 48-gon.

Use π = 3.14

Measure of central =_____________

b. Find the radius by solving this for r.

arclength = (measure of central 360) 2 r

Radius = _____________

c. Use Area = r² to calculate the area of the circle.

Area =_______________

d. Use Circumference = 2 r to calculate the circumference of this

circle.

Circumference =______________

Support pillar

Measure from here

To here



4. Now go to the Wave Swinger and complete the grid.

Number of sides in

polygon

Measure of one

central

Measure of one

interior

Measure of one

exterior

5. Complete a grid for the Enterprise. The radius of the Enterprise is 8.5 meters.

Use this to find the arclength between each pair of consecutive radii. A car serves

as an arc.

Number of sides in

polygon

Measure of one

central

Measure of one

exterior

Arc length between

radii

Use this regular polygon



6. Complete a grid for the Apollo 2000

Number of sides in

polygon

Measure of one central

Measure of one exterior

Why do the angle measures in this regular polygon

have repeating decimals?

What makes this polygon different from the previous

rides you’ve studied?

Explain here:

The connecting arms create the polygon you are

studying. Make a sketch of it in this space.



Polygon Grid Activity Please complete this activity prior to attending Dorney Park Thrill U. day.

# sides

# angles

# ’s

Sum of

measures of

interior ’s in

polygon

Measure of 1

interior in

regular polygon

Measure of 1

exterior in

regular

polygon

Sum of

measures of

exterior ’s in

polygon

3

5

6

7

8

9

10

12

14

16

20

Using the pattern above, fill in the rest of the table:

43

38

23

170

172.5

175

7.2

4



If the number of sides is expressed as N, then write a formula for the number of triangles, the

sum of the measures of the interior angles in a polygon, the measure of 1 interior angle in a

regular polygon, the measure of 1 exterior angle in a regular polygon, and the sum of the

measures of the exterior angles in a polygon.

# sides

# angles

# ’s

Sum of measures

of interior ’s in

polygon

Measure of 1

interior in regular

polygon

Measure of 1

exterior in

regular polygon

Sum of measures

of exterior ’s in

polygon

N

The Ferris Wheel


Information Sheet

topics: similar triangles

circumference

velocity

perimeter and area of a polygon

angle measures

objectives: to estimate the diameter of a circle

to determine the speed that a rider moves on the Ferris Wheel.

to calculate the perimeter, area and other measurements pertinent to

polygons.

equipment: activity sheets

pencil

calculator

stop watch

ruler and string, or protractor with plumb and straw


Students will need to estimate the diameter of the wheel of the Ferris

Wheel. In order to determine this distance, students may use similar

triangles or trigonometry. This is an important part of this activity. The

activity cannot be completed without determining these diameter.

If using the similar triangle method on the Ferris Wheel, a 6 inch ruler

works better than a 12 inch ruler.

You may wish to direct your students to give all linear measures in feet,

and square measures in square feet.

Special note: To successfully complete this activity, students are not

required to ride the Ferris Wheel.

The Ferris Wheel


d

h B

line of sight

string h = height (ground to eye level)

C

A

C1

ruler

B1

ground level

PART 1:

The thrill rating of this ride is __________.

1. Estimate the diameter of the wheel. This can be done using similar triangles or

trigonometry.

A. Similar Triangle Method: Stand at the curb across from the ride entrance. This

point is approximately 47 feet from the wheel. Hold your ruler up as if to

measure the diameter of the wheel. Stretch the string from the bottom of the ruler

to your eye and measure this length. Use proportional sides of similar triangles to

estimate the diameter of the wheel.

(Note that the length of side B1C1 is greater than the diameter of the wheel.)

B. Trigonometry Method: Stand at the curb across from the ride entrance. This

point is approximately 47 feet from the wheel. Attach the plumb to the

protractor at the vertex mark of the protractor. Attach the straw to the straight

edge of the protractor. Use this device to determine the angle of elevation. Use

the appropriate trigonometric function to estimate the diameter of the wheel.

2. Find the circumference of the wheel.

10ft.8in.

The Ferris Wheel


3. Determine the average time that it takes the wheel to make one complete revolution.

Time at least three revolutions and find the average.

4. Using the results of #2 and #3, determine the velocity (speed) in feet per second that a

rider moves when the wheel is in motion.

5. If the diameter were half as large, but the time to complete one revolution remains the

same, will the rider move faster or slower on the new Ferris Wheel? How much

faster or slower?

The Ferris Wheel


PART 2:

1. Determine the average number of revolutions per minute (rpm) that the wheel makes.

2. Determine the angular velocity, , in radians per second.

= rpm's x 2

60

3. Multiply the angular velocity by the radius of the wheel.

4. Compare this result with the answer to #4 of PART 1. What did you find? Why do

you think that you got your result?

The Ferris Wheel


PART 3:

Look at the wheel. You should see an inscribed polygon.

1. Count the number of sides. _______________

2. What special name applies to this polygon? _______________

3. Is this polygon regular? ______________

4. Find the measure of AOB. __________

5. Find the measure of OAB. __________

6. Find the measure of ABC. __________

7. Radius OA = __________.

8. Apothem OZ = __________.

9. Find the perimeter of the polygon. __________

10. Find the area of the polygon. __________

A

Z B C

X

D

O

The Ferris Wheel


11. DCX is called _______________.

12. Find the measure of DCX. __________

13. How does mAOB compare to mDCX? ______________________________

14. What is the sum of the measures of the angles of the polygon? __________

15. What is the sum of the measures of the exterior angles (one at each vertex) of the

polygon? __________

The Sea Dragon


Information Sheet

topics: Pythagorean theorem

equilateral and isosceles triangles

congruent triangles

corresponding parts of triangles

right triangle trigonometry

objective: to find the lengths of the stationary beams

to find the degree measure of all of the angles formed by the support

beams


pencil

calculator (scientific or better)


The best view of the structure of Sea Dragon is at the exit side of the ride.

The Sea Dragon


The stationary support beams of the Sea Dragon form numerous triangles. Your

job is to calculate the measures of the unknown beams as well as all of the angles within

the stationary support structure.

Below is a drawing of the support structure as viewed from the exit side of Sea

Dragon. The solid lines represent actual beams and the dotted lines represent constructed

lines needed for relationships and calculations.

B

E

A C D 1. ABC is an equilateral triangle with side 46 ft.

An equilateral triangle can also be classified as an triangle.

a. AB = BC = AC =

mBAC = º mBCA = º mABC = º

NOTE: BD is an altitude of equilateral triangle ABC.

Therefore BDA BDC because .

mBDA = mBDC = º.

Consequently, ABD and CBD are triangles.

The Sea Dragon


b. Since ABC is equilateral and BD is an altitude,

AD = and CD = .

Explain:

c. We can now calculate the length of BD using the Pythagorean Theorem on

either ABD or CBD.

(altitude)² + (base)² = (hypotenuse)²

What information have we found that allows us to use the Pythagorean

Theorem?

Using either ABD or CBD, put the mathematical names of the sides into

the Pythagorean Theorem.

( )² + ( )² = ( )²

Replace the names of the sides with their measures from #1a and #1b and

calculate the length of BD to two decimal places, showing all of your work.

BD =

2. AED is a triangle.

a. mEDA = º and ED = 20 ft.

From #1b we know AD = . How can we calculate AE ?

Doing this calculation yields AE = .

The Sea Dragon


b. AED CED because .

Therefore CE = because

.

c. To calculate the measure of EAD we can use a trigonometric ratio.

Fill in the blanks for these ratios using side names from AED.

sin(mEAD) = cos(mEAD) =

tan(mEAD) =

Replace the side names with their measures.


tan(mEAD) =

Find decimal values for the ratios above.


tan(mEAD) =

Use an inverse function on your calculator to find the angle measure.

Fill in the blanks with the appropriate decimal values from above.

Sin-1 = mEAD Cos-1 = mEAD

Tan-1 = mEAD

mEAD = º (to the nearest degree)

The Sea Dragon


d. Show two different ways that you can now calculate mECD.

mECD = º (to the nearest degree)

e. Explain how you will find mAED and mCED.

mAED = º mCED = º

3. AEB CEB because

a. Use your answer from #1c and information given in #2a to calculate BE.

(Show your work.)

BE =

b. Based on information found in #1a and #2c show how can we find mBAE.

(Show your work.)

mBAE = º

The degree measure of BCE = because

The Sea Dragon


c. How can we determine the measures of ABE and CBE?

Find mABE and mCBE.

mABE = º mCBE = º

d. How can we determine the measures of BEA and BEC?

Find mBEA and mBEC.

mBEA = º m BEC = º

Steel Force


Information Sheet

topics: angle of depression

right triangle trigonometry

slope

objective: to approximate and compare the measures of the angles of depression and the

slopes of the first and second hills using right triangle trigonometry and the

definition of slope

equipment: activity sheets and pencils

calculator

ruler or tape measure


Please remind students that their answers should include appropriate units of

measure.

Steel Force


PART 1:

Approximate the angle of depression of the first hill. The vertical drop is given. The length of

the hill can be estimated by counting the number of vertical supports on the track.

1. Find the number of supports from the top of first hill to the point where it enters the

tunnel. (Hint: you can get a good view of these supports while standing in line at

Thunderhawk.)

The number of supports is _______________.

2. Calculate the length of the first hill.

a. From the answer in #1 above, subtract 2. ____________

b. To find the approximate length of the first hill, multiply the number in #2a by 24 feet

(distance between supports). This is the approximate length of the first hill.

The length of the first hill is __________________________________.

3. Use trigonometry to find mBAC.

mBAC is _______________.

vertical drop

(200 feet)

Count vertical supports to

find this distance.

A (top of first hill)

C (tunnel entrance)

angle of depression

B

Steel Force


4. Find the measure of the angle of depression.

The measure of the angle of depression of the first hill is _______________.

PART 2:

Approximate and compare the angle of depression of the second hill to that of the first hill.

1. Find the horizontal distance between the top of the second hill and the point where the

track crosses the water (EF). Stand on the midway at a point directly below the top of the

second hill. Walk on that midway in the direction of Thunder Creek Speedway. Count

the number of paces you take until you reach the point where the track crosses the water.

The number of paces is _______________.

2. Measure the length of each pace. (Use the average of three paces to get an accurate

measurement.)

The length of one pace is _______________.

3. Calculate the horizontal distance (EF).

The horizontal distance between the top of the hill and the point where the track crosses

the water is _______________.

Pace off this distance.

Count the number of

supports to get this length.

point where the track

crosses the water

D

E F

top of the second hill

Steel Force


4. Count the number of supports from the highest point on the second hill to the point where

the track crosses the water.

The number of supports is _______________.

5. Calculate the length of the second hill.

a. From the answer in #4 above, subtract 2. ____________

b. To find the approximate length of the second hill, multiply the number in #5a by 24 feet

(distance between supports). This is the approximate length of the second hill.

The length of the second hill is ______________.

6. Use trigonometry to calculate mEDF.

mEDF is _______________.

7. Find the measure of the angle of depression of the second hill.

The measure of the angle of depression of the second hill is _______________.

8. Describe the differences between the angles of depression for the two hills on Steel

Force.

9. Calculate the slope of each hill.

The slope of the first hill is ________________.

The slope of the second hill is ______________.

10. Describe the differences in the slopes of the two hills.

White Water Landing – The Bridge


Information Sheet

topics: circle circumference as related to radius

linear and degree measure of a circle

simple right triangle trigonometry

objective: to determine the relative location of the center of the circle

to appreciate the relationship between the circumference and radius of a circle


sextant

pencil

calculator


Students must construct a 20 ft. measuring string before arriving at the

park. The string must have markings every 6 inches (½ ft.) and weighted on one

end using fishing sinkers, washers, nuts, or the like. The 6-inch markings should

be measured beginning from the outside of the weight, where the string is

attached.

Students must also construct a simple sextant in order to measure angles

vertically. Materials for the sextant: protractor with one-degree markings, a

straw, fishing line, and a weight. Tie the fishing line to the protractor at the center

of the straight edge (or through the hole if available), tape the straw to the straight

edge of the protractor, tie the weight to the other end of the line so it can swing

freely from the center of the protractor.

Please note: Due to safety precautions, students must ride this ride to gain access to the bridge.

Remind students that all equipment must be secured when riding the ride.



Goal: Determine the location of the center of the circle, formed in part by the arched support of

the center section of the bridge at White Water Landing. (See photo below).

ARCHED

SUPPORT

1. The arched support as labeled above is an arc of a circle.

a. How many degrees are in a circle? ______

b. If the arched support has an arc measure of 12 degrees, what fractional part of the

circle does the arched support represent? (Give your answer as a fraction and as a

repeating decimal.)

fraction______ decimal______________

2. Find the distance (linear measure) from the top center of the arc to the water.

To calculate this distance, you will be using your sextant.



Point X is a point along the fence just to the right of the lamppost (with the red dot).

Point T is a point on the top of the arched support aligned with the middle of the top of the sign.

Point B is a point at the bottom of the bridge below the middle of the bottom of the sign.

Point W (marked in red) is a point on the boat track, at water level, directly below the bridge.

T

104.87

B X

W

a. Hold the sextant level with the top of the fence, look through the eyepiece (straw),

and locate point B. (A team member must make sure the fishing line is not

swinging and is aligned with the 90 mark on the protractor part of the sextant,

while you are looking at point B.) Raise the front of the sextant from the

horizontal to locate point T at the same position in the eye piece where point B

was. Have the team member read the degree measure on the sextant (to the

nearest 1/2-degree) when you have T in the correct position.

The measure of BXT = _______.

(Make sure the line has stopped swinging before reading mBXT.)

b. Following the same procedure as in #2a, locate point B (recall that the line must

be still and aligned with 90), lower the front of the sextant from the horizontal to

locate point W at the same position in the eye piece where point B was. The

measure of BXW = _______.

You will be using the trigonometric function tangent (TAN) on your calculator to

find BT and BW.

side opposite the angle

Given: TAN (angle) = side adjacent to the angle

and BX = 104.87 ft., the distance from the fence to the bridge.



c. Find BT using the tangent.

TAN (mBXT) = BX

BT

TAN (mBXT) = .87.104 ft

BT

(Use the measure of BXT from #2a to continue.)

TAN (mBXT) = .87.104 ft

BT

BT = (104.87 ft.) TAN (mBXT)

BT = _________ft.

d. Calculate BW using the same procedure as in #2c.

TAN (mBXW) = .87.104 ft

BW

(Use the measure of BXW from #2b to continue.)

TAN (mBXW) = .87.104 ft

BW

BW = (104.87 ft.) TAN (mBXW)

BW = ________ft.

Using the information from #2c and #2d, how can you determine the distance

from the top center of the arc to the

water?__________________________________________________________.

TW (arc to the water) = ___________ft.

3. Find the circumference of the circle formed by the arc of the arched support.

a. In #1, we determined the fractional part of the circle to be_____________ and the

decimal part of the circle that the arc represented to be________________.



b. Determine the linear measure of the arc using your measuring string. Measure

along the top of the arc from the red dot to either end of the arc. (Measure to the

nearest 1/4 of a foot.)

length of ½ of the arc = _________ ft.

c. The entire length of the arc is approximately _________ft.

d. Based on measurements from #2d and #3c and without doing any calculations,

where would you hypothesize (educated guess) the center of the circle to be

located? (above the water, underwater but above the ground, or underground)

_______________________________ __

Explain the reasoning for your conclusion_______________________________

_________________________________________________________________

_________________________________________________________________.

e. Using the information in #3a and #3c, the circumference of the circle is

__________ft.

Show work here:

4. Find the radius of the circle.

a. In #2d, we found the distance from the top center of the arc to the water line to be

_________ft. and the depth of the pool is 2.5 ft.

b. The circumference of a circle in terms of the radius is calculated by the formula

C = ___________.

In #3e, we found the circumference to be ________ft.

How can we calculate the radius based on the circumference? _______________

_________________________________________________ ___.

c. Doing this calculation yields a radius of _________ft.



5. Based on your calculation in #4c and measurements in #4a, where is the center of the

circle located? (above the water, underwater but above the ground, or underground)

_________________________________________________________________.

Explain your answer:

_________________________________________________________

_______________________________________________________________________.

a. Does this answer match your hypothesis in #3d? _____________

b. If not, why do you think your guess was wrong?

____________________________________

___________________________________________________________ __.

A Geometric Walk

KUTZTOWN UNIVERSITY OF PENNSYLVANIA

Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2

91

Information Sheet

topics: recognition of geometric shapes

area of square, trapezoid, triangle, circle, rectangle

circumference of a circle

volume

ratio

objectives: to use appropriate formulas for area, volume and circumference

to identify geometric shapes in a real world situation


pencil

50-foot tape measure

basic calculator


This is a good activity for working in pairs. Students need to record and measure

as a team.

When working with the outer rim of bricks, use two rows of bricks as a border. In

other words, the identical 4 geometric shapes do not overlap.

A Geometric Walk



92

As you walk through the main gate of the Park, you will enter the plaza situated between

the gate and the Antique Carrousel.

PART 1:

Looking down as you walk into the plaza, you will see four identical geometric shapes

each of which look like Figure 1 below. The shaded portions indicate the location of

bricks. Do not include the geometric shapes that have a center planter.

Figure 1 - The Plaza Shape

1. Name the different geometric shapes that can be found in this section of the plaza

walkway.

2. Find the areas of the following geometric figures found on the plaza walkway. Do

not include the outer rim of bricks in your calculations.

a. Area of rectangle ABCD

b. Area of trapezoid DEFC

c. Area of square EFGH

d. Area of triangle ABD

A B

D C

H G

E F

A Geometric Walk



93

3. Knowing the areas of the geometric figures in question #2, answer the following:

a. Use the fact that the area of a geometric figure is equal to the sum of the areas of

its parts to show that the area of rectangle ABCD is equal to the sum of the areas

of its parts.

b. Remember that there are four geometric shapes on the entire plaza. What is the

total area of all the trapezoids in all four figures? Check your answer by using the

areas of the rectangle and square.

4. The rectangular shaped figures on the walkway are surrounded by and include

rectangular and square bricks.

a. Find the area of one of the rectangles that includes the outer border of bricks

(Figure 1).

b. Find the ratio of the number of rectangular bricks to the number of square bricks

surrounding and included in one of the four rectangles in the plaza.

A Geometric Walk



94

c. In #2, the areas of the rectangle ABCD and the square EFGH were found. Use

these areas to find the total area of the shaded regions in Figure 1.

d. Use the ratio found in #4b and your answer in #4c to determine the area covered

by the rectangular bricks in the shaded regions in Figure 1.

e. Use the ratio found in #4b and your answer in #4c to determine the area covered

by the square bricks in the shaded regions in Figure 1.

f. Find the area covered by the square bricks by using the measurements for the size

of the brick. Also find the area covered by the rectangular bricks using the

measurements for the size of the brick.

g. Compare the answers from parts d, e and f above. Reflect on your

answers.

A Geometric Walk



95

PART 2 (probability):

What is the percent probability of a bean bag being drop from a helicopter landing on one of the

following patterns? Hint: use the following formula P(target area) = area of target / area of total

a. The square EFGH

b. The trapezoid ABGH

c. Any trapezoid

d. The brown brick boarder of the plaza shape

PART 3:

If you walk further down the walkway past the Antique Carrousel, you will see several

rectangular planters containing trees and mulch on the main Midway. Look at the planters in line

with the fountain at the center of the Midway.

1. Find the area of the first rectangular planter that is not a square.

Use the outside edge of the concrete to measure the sides. Do not include any bricks

surrounding the planter.

A Geometric Walk

KUTZTOWN UNIVERSITY OF PENNSYLVANIA 96

2. Find the area of the section of the rectangular bed that contains mulch. Explain the

procedure to find the area covered by only mulch.

(Assume that the area to be covered is clear of all electrical boxes, trees, etc.)

3. Given that mulch can be purchased by the cubic yard, estimate the number of cubic

yards of mulch that would be needed to fill one rectangular bed. (One cubic yard of

mulch covers approximately 80 to 100 square feet if spread 3 inches deep.)

Extension for Experts:

4. If the mulch can only be ordered by the bag, how many bags would be

needed? One bag of mulch contains 3 cubic feet of mulch.

A Geometric Walk


PART 4:

In the middle of the main Midway, you will find a circular fountain filled with water. The

fountain’s structure contains a middle circular concrete island. The six outer segments are flower

beds.

Figure 2 - The Fountain

1. Find the area and circumference of the circle formed by the outer edge of the fountain

given that the radius of the circle is 97 inches.

Area: _______ sq. inches

Circumference: _______ inches

2. Find the area (i.e. the surface area) of the part of the fountain that holds water. The

middle circular island has a radius of 31 inches. The distance from the edge of the

circular island to the interior wall of the fountain is 54 inches. This geometric shape

looks like a doughnut.

3. The usual depth of the water in the fountain is 8.5 inches. Find the volume of

water needed to fill the fountain to this depth.

Polygons in the Park


Information Sheet

topics: classification of a wide variety of polygons

identification of convex, concave, regular and non-regular polygons

objectives: to recognize and name polygons inherent in the Park

to accurately classify triangles and quadrilaterals

to recognize characteristics of polygons (convex, concave, regular, etc.)

to order the measures of areas of triangles and quadrilaterals

to sketch polygons

equipment: activity sheets and pencils


Please keep in mind that this activity is not an activity on measurement. Students

will be unable to obtain measurements of angles and lengths of sides for most of

the polygons. Their answers for columns 3, 4, and 5 on the “Answer Grid” will

be based on observations and reasoning skills.

For questions with multiple answers (i.e. #4 and #13), please allow for different

arrangements of the answers.

For responses about the polygon in PART 1, #11, students should respond as

though the hole were not there.



PART 1:

The goal of this activity is to be able to accurately describe many different polygons that can be

found in Dorney Park. (Please read through PART 2 before you start PART 1.)

Place all of your answers on the grid that is provided at the end of this activity.

For each of the following,

a. identify (by its mathematical name) the polygon that is being described,

(If you see a polygonal region, identify the polygon that describes its boundary.)

b. draw a sketch of what you see;

c. identify it as regular or not regular; and

d. identify it as convex or concave.

Your answers will be terms such as: pentagon, hexagon, decagon, dodecagon, icosagon, 36-gon,

triangle, quadrilateral, etc. If you find a triangle or a quadrilateral, list as many names for it as

you can: isosceles, equilateral, right, acute, obtuse, rectangle, square, parallelogram, trapezoid,

or rhombus.

1. The base (bottom) of a car on the Ferris Wheel has this polygonal shape.

2. The metal bars that join the supports between the swings on Charlie Brown's Swing form

this polygonal shape.

3. At Musik Express, each letter of the sentence "MIT MUSIK GEHT ALLES

BESSER" is written on one of these.

4. The center structure on the wheel of Musik Express is comprised of these three

polygonal regions.

5. The base of the center support of the Scrambler has this shape.

6. The roofs on the cars on the Enterprise, laid end to end, form this polygon.

7. At the top of the Dominator, in a plane parallel to the ground, there is a polygon that

"connects" the three vertical supports. Name the polygon.

8. The roof of Patio Pizza is decorated with black and white polygons of this type.

9. This polygon can be viewed on the ground (made from red/brown bricks) and is the

smallest polygon that surrounds the Antique Carrousel.



10. Above the windows at the SUBWAY sandwich shop, there are polygonal signs that say

"PAY HERE" or "ORDER HERE." What is the shape of each sign?

11. On the center support of the Scrambler, this polygonal region has a circular hole in it.

(When working with this polygon, ignore the hole.)

12. At the Funzone Arcade, there are two different shapes that make up the three-piece

emblem above the neon "FUNZONE" sign. Identify both polygons.

13. The arms that hold the cars on Apollo 2000 are connected to each other at the top of the

ride. In a plane parallel to the ground, name the polygon that is formed by the

connections of all the arms of the cars.

14. The perimeter of the pavilion in Camp Snoopy has this shape.

PART 2:

1. Arrange the areas of all of the regions enclosed by triangles that were found in PART 1

from smallest to largest. Do this by listing the numbers of the objects. (Do you really

need any formulas or computations of areas?)

2. Arrange the areas of all of the regions enclosed by quadrilaterals that were found in

PART 1 from smallest to largest. Do this by listing the numbers of the objects. (Do you

really need any formulas or computations of areas?)



Answer Grid for Polygons in the Park

Problem number and

Attraction

Names for the

polygons

Sketch Regular or

not regular?

Concave or

convex?

1. Ferris Wheel

2. Charlie Brown’s

Swing

3. Musik Express

(letters)

4a. Musik Express

(center)

4b. Musik Express

(center)

4c. Musik Express

(center)

5. Scrambler (base)

6. Enterprise

7. Dominator

8. Patio Pizza



Problem number and

Attraction

Names for the

polygons

Sketch Regular or

not regular?

Concave or

convex?

9. Antique Carrousel

10. Subway

11. Scrambler (hole)

12a. Funzone Arcade

12b. Funzone Arcade

13. Apollo 2000

14. Pavilion at Camp

Snoopy

Rides That are Inscribed


Information Sheet

topics: properties of regular polygons (interior & exterior angles)

area of regular polygon

central angles

area of circle

area of sectors

objectives: to use properties of regular polygons

to use formulas for determining measures of interior and exterior angles

to calculate lengths and areas associated with polygons inscribed in circles


pencil

basic calculator

notes to teacher:

This activity contains three parts.

PART 1 – Chart form investigation of interior and exterior angles.

PART 2 – Question and answer investigation of regular polygons.

PART 3 – Area investigation of inscribed polygons.

PART 1: Regular Polygons – Interior & Exterior Angles Chart

Attendance at the Park is needed to complete the Name of Ride column of the chart.

If time is a factor, the other columns may be completed/calculated at a later time.

http://mathforum.org/dr.math/ is a good source for the polygon names for 11 and

up, if you are not going to use 11-gon, 12-gon, etc.

PART 3: #1, 2, & 3

The circle provided has divisions of 5.

http://mathforum.org/dr.math/



PART 1: Regular Polygons – Interior & Exterior Angles Chart

Walk around the park. Look carefully at each ride. You should start to notice that most of the rides

listed are circular in nature. An actual or visualized polygon is inscribed within the circle. For example

an actual polygon exists on the Ferris Wheel. The vertices of the polygon are located at the ends of the

spokes. The Tilt-A-Whirl’s polygon is visualized by joining segments from each seat’s pivot point.

Examples of how the rides may appear (bird’s eye view) as inscribed polygons:

KEY: Actual structure Visualized structure

1. Why do you think most of the rides have polygons with an even number of sides?

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

Instructions for Regular Polygons – Interior & Exterior Angles chart.

a. Determine where the 10 rides listed below fit into the chart.

Ride/Structure list: Antique Carrousel

Apollo 2000

Dominator

Ferris Wheel

Monster

Musik Express

Pavilion near Fossil Find and Snoopy Bounce

Scrambler (configuration of seating area)

Tilt-A-Whirl

Wave Swinger.

b. Fill in the Name of Polygon column. You may need your math book for this.

c. Place the # of Vertices in that column.

d. Complete the Measure of each Exterior column, using your information from # of

Vertices and Sum of the measures of Exterior ’s columns.

e. Use the information in your chart to complete the last two columns.



Regular Polygons – Interior & Exterior Angles

Name of Ride

#

Sides

Name

of Polygon

# of

Vertices

Sum of the

measures

of

Exterior

’s

Measure of

each

Exterior

Sum of the

measures

of

Interior

+

Exterior

Measure of

each

Interior

Sum of the

measures

of

Interior

’s

3 360 180

4 360 180

* 5 360 180

6 360 180

7 360 180

8 360 180

* 9 360 180

* 10 360 180

* 11 360 180

12 360 180

* 13 360 180

14 360 180

* 15 360 180

16 360 180

* 17 360 180

18 360 180

* 19 360 180

20 360 180

* See if you can find a ride or structure to match the number of sides.



PART 2:

Answer the following questions about polygons:

1. What makes a polygon regular?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

2. Use the figures below. Draw the diagonals from one vertex to help complete the

following chart:

Number of sides

3

4

5

6

7

8

9

n

Number of diagonals

from one vertex

Number of triangles

created by diagonals

Sum of the m’s in a

Sum of the measures of

the interior’s in the

polygon

3. Write a formula for the sum of the measures of the interior angles of a convex polygon with n

sides.

________________________________________________________________________

4. Write a formula for the measure of each interior angle of a regular polygon with n sides.

________________________________________________________________________

5. What is the sum of the measures of the exterior angles of a convex polygon (one angle at

each vertex)?

________________________________________________________________________

6. Write a formula for the measure of each exterior angle of a regular polygon with n sides.

________________________________________________________________________



Select True or False. Support your decision with a brief explanation.

7. The sum of the measures of the interior angles of a polygon is always a multiple of 180.

True or False Explanation: _____________________________________________

_____________________________________________________________________

_____________________________________________________________________

8. There is a regular polygon in which the sum of the measures of the interior angles is 310.


_____________________________________________________________________

_____________________________________________________________________

9. As the number of sides of a regular polygon increase, the measure of each interior angle

will decrease.


_____________________________________________________________________

_____________________________________________________________________

10. As the number of sides of a regular polygon increase, the sum of its exterior angle

measures will also increase.


_____________________________________________________________________

_____________________________________________________________________

11. Each exterior angle of a regular hexagon is acute.


_____________________________________________________________________

_____________________________________________________________________



PART 3:

In the circles provided, accurately inscribe the regular polygon represented by each of the

following rides: Dominator (completed for you), Scrambler (seat configuration), and Monster.

Draw at least one central angle and one apothem. For each, calculate the length of the apothem,

perimeter of the polygon, measure of one central angle, area of one triangle, and area of the

polygon.

1. Dominator: Draw regular polygon ABC inscribed in circle O.

Given: AB = 40 feet Find the following:

2. Scrambler (seat configuration): Draw regular polygon ABCD inscribed in circle O.

Given: diameter = 260 inches Find the following:

A. apothem of polygon ABCD = __________

B. perimeter of polygon ABCD = __________

C. measure of central AOB = __________

D. area of AOB = __________

E. area of polygon ABCD = __________

EXTENSION: F. area of sector AOB = __________

G. area of region bounded by

AB and arc AB = __________

H. difference between the area of

polygon ABCD and circle O = __________

O

C

a B A

A. apothem a of polygon ABC = __________

B. perimeter of polygon ABC = __________


D. area of AOB = _______

E. area of polygon ABC = __________

EXTENSION:

F. area of sector AOB = __________


AB and arc AB = __________


polygon ABC and circle O = __________



3. Monster: Draw regular polygon ABCDEF inscribed in circle O.

Given: diameter = 42 feet Find the following:

A. apothem of polygon ABCDEF = __________

B. perimeter of polygon ABCDEF = __________


D. area of AOB = __________

E. area of polygon ABCDEF = __________

EXTENSION:

F. area of sector AOB = __________


AB and arc AB = __________


polygon ABCDEF and circle O = __________

The Fish Pond

KUTZTOWN UNIVERSITY OF PENNSYLANIA

Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander

110

Information Sheet

Topics: Circumference

Area

Volume

Trigonometry

Objectives: To use the circumference of a circle to determine its radius.

To calculate the area and volume of a cylinder given its dimensions.

To calculate the height of an object using trigonometric ratios.

Equipment: Pencil

Calculator

Activity sheets

Tape measure

String

Inclinometer

The Fish Pond



111

The historic fish pond has been here since the 1920s. It is located near the entrance to Gasoline

Alley.

Part I:

1. Find the circumference of the pond.

a. Walk around the fish pond to find the gate. It is near the hedge in the direction of

Steel Force.

b. Measure around the inside of the cement base of the pond from middle of post to

middle of post on the gate. Use a string to do so, then measure length with a

measuring tape.

c. Measure one section on either side of the gate. Both are congruent, so we only

need to measure once. Again, use the string and go from post to post around the

inside of the cement base.

d. The remaining sections of fence are congruent. Take a moment to walk around

the pond and count the number of remaining fence sections.

e. Measure one of these sections and record your results here.

f. Calculate the circumference of the fish pond by adding all of your measurements.

Remember to include all sections of fence and the gate.

The Fish Pond



112

2. Calculate the area and volume of the pond.

a. Recall that circumference of a circle is found using the formula C = 2 r. Using

the circumference, find the radius of the pond.

b. In order to calculate the area of the fish pond, we must take into consideration the

fact that there is an island in the center. Given that the radius of the island is 10

feet, calculate the area of the pond. Recall that A = r2.

c. Given that the depth of the pond measures 4 feet, calculate the volume of the

pond. This will tell us the maximum amount of water that it could be contained

within the pond. Don’t forget about the island in the center!

3. Find the height of the center tower.

a. Using the top railing of the fence as your starting point, steady the inclinometer.

Tilt it upward until the top of the tower is in your sight. Have a teammate read

the angle of inclination.

The Fish Pond



113

b. Using the angle of inclination and the radius of the pond, calculate the height of

the tower from the railing upward. Use the diagram as a reference.

c. Because we started at the railing, instead of the ground, this is an incomplete

height. You must now find the height from the ground to the top of the

inclinometer as it rests on the top railing.

d. Adding these two lengths together will give you the height of the tower.

Radius

Height

Angle

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