The Relationship Between Circumference and Diameter

Jennifer M. Seay

Wicomico Middle School

Jseay@wcboe.org

Grade 8 Pre-Algebra and Algebra 1

 

Topics:

This lesson is an introduction to finding the formula for the circumference of a circle. It is understood that the students have experience in measurement, along with basic mathematical operations.

 

Purpose:

The purpose of the lesson is to allow students to discover the meaning of the value of pi and apply that value to finding the circumference of a circle.

 

Content Standards (Maryland Core Learning Goals) and Objectives:

1.1.1        Given a…geometric representation or description of a pattern or functional relationship, the student will give a verbal description or predict the next term or a specific term in a pattern or functional relationship.

1.1.2        The student will represent patterns and/or functional relationships in a table, as a graph, and/or by mathematical expression.

1.2.5        Given a formula, students will substitute values, solve and interpret solutions in the context of a problem.

 

Materials Needed:

  1. Five round objects (i.e. top of teacher’s stool, film canister, paper plate, trash can, cup)
  2. Five pieces of paper labeled 1 to 5 for creating “stations.”
  3. String (about 1 ½ meters in length)
  4. Scissors (1 pair)
  5. Meter Sticks (5)

 

Time Required:

Lesson Procedure:

10 to 15 minutes
  1. Begin the class by asking the students how they would find the perimeter of a given rectangular object in the room. Discuss how they would go about measuring the sides and finding the total perimeter.
  2. Talk about the formulas for perimeter of regular polygons. What happens as the number of sides in the polygon increases? (The number that you multiply by increases).
  3. Define a circle as a polygon with an infinite number of sides. How would one go about finding the “perimeter” of this type of polygon? Take suggestions and write them on the board.

15 to 20 minutes
  1. Hand out a table to each student for recording values.  Explain that they are to take the measurements first, then we will create a list of class data. Finally, they will calculate the requested values after gathering all data.
  2. Assign each student to a group, numbered 1 to 5. Give one person in the group a meter stick and a piece of string.
  3. The students should start at the station that matches their team number.  They will have two minutes to measure the circumference and diameter of the circle at their station.
  4. After two minutes call “switch,” and the students should move to the next station.
  5. When they have finished measuring at all stations, they should sit back in their seats.

10 to 15 minutes
  1. Record class data on a table at the front of the classroom.  Have the students discuss (and make a list of) reasons that the numbers they got were not exactly the same from group to group.
  2. Discuss a way that the discrepancies in data can be remedied. Suggestions should include taking the mean or median of the values, taking more trials of measurements, etc.
  3. Students should, in their groups, come to consensus about what data values they will use as a class.
  4. Have the students record the class values and complete the rest of the chart.
  5. The students should plot the diameter and circumference on the graph paper. First, discuss which variable goes on which axis.

 
  1. Discuss the values that the students got when they divided circumference by diameter. Do they see any patterns in one of the operations? They should see that the quotient of C and D is always approximately equal to 3. They should also see that the slope of the line is also about three.
  2. Introduce the value of pi as a rational number that represents the ratio between circumference and diameter.

5 minutes

 
  1. Introduce the formula for circumference of a circle to the students. Have them find the circumferences of the circles using the formula, then have them find the circumferences of other circles (predicting, using the model) with different diameters.

Closure
  1. Review what was done during the class. Remind the students that Pi is an irrational number that represents the ratio between circumference and diameter, and that the relationship between circumference and diameter is a linear one.

 

Assessment and Evaluation
Students will be informally assessed during the lesson, based on their accuracy in measuring the circumferences and diameters of the given circles. There will be a small quiz at the end of the week that will test their ability to remember and use the formula for the circumference of a circle, and the students will be formally assessed during later tests as well. 
 
Explorations and Extensions
A later lesson will explore the relationship between radius and area of a circle. This will only be done after the students have had some time to discover relationships beyond linear ones.
 
Comments from the Author:
This is a very effective lesson for getting kids to really understand the formula for circumference of a circle. They not only discover the formula for themselves, but they understand the meaning of the value of pi, something that was not thoroughly explained to many of us when we were young. I have done the lesson before, but not with the graphing piece, so I am curious to see how it turns out this year. 
 
For more about Pi see the Dr. Math FAQ:
  http://mathforum.org/dr.math/faq/faq.pi.html   

 


Ptolemy

(c. 150 AD)

3.1416

Tsu Ch'ung Chi

(430-501 AD)

355/113

al-Khwarizmi

(c. 800 )

3.1416

al-Kashi

(c. 1430)

14 places

Viète

(1540-1603)

9 places

Roomen

(1561-1615)

17 places

Van Ceulen

(c. 1600)

35 places

1699:

Sharp used Gregory's result to get 71 correct digits

1701:

Machin used an improvement to get 100 digits and the following used his methods:

1719:

de Lagny found 112 correct digits

1789:

Vega got 126 places and in 1794 got 136

1841:

Rutherford calculated 152 digits and in 1853 got 440

1873:

Shanks calculated 707 places of which 527 were correct

 

3.
141592653589793238462643383279502884197169399375105820974944
592307816406286208998628034825342117067982148086513282306647
093844609550582231725359408128481117450284102701938521105559
644622948954930381964428810975665933446128475648233786783165
271201909145648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367892590360
011330530548820466521384146951941511609433057270365759591953
092186117381932611793105118548074462379962749567351885752724
891227938183011949129833673362440656643086021394946395224737
190702179860943702770539217176293176752384674818467669405132
000568127145263560827785771342757789609173637178721468440901
224953430146549585371050792279689258923542019956112129021960
864034418159813629774771309960518707211349999998372978049951
059731732816096318595024459455346908302642522308253344685035
261931188171010003137838752886587533208381420617177669147303
598253490428755468731159562863882353787593751957781857780532
171226806613001927876611195909216420198938095257201065485863
278865936153381827968230301952035301852968995773622599413891
249721775283479131515574857242454150695950829533116861727855
889075098381754637464939319255060400927701671139009848824012
858361603563707660104710181942955596198946767837449448255379
774726847104047534646208046684259069491293313677028989152104
752162056966024058038150193511253382430035587640247496473263
914199272604269922796782354781636009341721641219924586315030
286182974555706749838505494588586926995690927210797509302955
321165344987202755960236480665499119881834797753566369807426
542527862551818417574672890977772793800081647060016145249192
173217214772350141441973568548161361157352552133475741849468
438523323907394143334547762416862518983569485562099219222184
272550254256887671790494601653466804988627232791786085784383
827967976681454100953883786360950680064225125205117392984896
084128488626945604241965285022210661186306744278622039194945
047123713786960956364371917287467764657573962413890865832645
995813390478027590099465764078951269468398352595709825822620
522489407726719478268482601476990902640136394437455305068203
496252451749399651431429809190659250937221696461515709858387
410597885959772975498930161753928468138268683868942774155991
855925245953959431049972524680845987273644695848653836736222

Pi was known by the Egyptians, who calculated it to be approximately (4/3)^4 which equals 3.1604. The earliest known reference to pi occurs in a Middle Kingdom papyrus scroll, written around 1650 BC by a scribe 
named Ahmes. He began the scroll with the words: "The Entrance Into the Knowledge of All Existing Things" and remarked in passing that he composed the scroll "in likeness to writings made of old." Toward the 
end of the scroll, which is composed of various mathematical problems and their solutions, the area of a circle is found using a rough sort of pi.
 
Around 200 BC, Archimedes of Syracuse found that pi is somewhere about 3.14 (in fractions; Greeks did not have decimals).  Pi (which is a letter in the Greek alphabet) was discovered by a Greek mathematician 
named Archimedes. Archimedes wrote a book called The Measurement of a Circle. In the book he states that Pi is a number between 3 10/71 and 3 1/7. He figured this out by taking a polygon with 96 sides and 
inscribing a circle inside the polygon. That was Archemedes' concept of Pi. 
 
New knowledge of Pi then bogged down until the 17th century. Pi was then called the Ludolphian number, after Ludolph van Ceulen, a German mathematician. The first person to use the Greek letter Pi for the 
number was William Jones, an English mathematician, who coined it in 1706.
 
In the 1800's people sat down for years on end to find the values of pi to about 1000 places. Imagine doing this by hand with no calculators. This has become a thing of the past, since the tedium that used to be done by hand is now done by computer. 
 
 
Facts About Pi
 - Pi is the ratio of the circumference of a circle to the diameter. 
 - 2 Pi in radians form is 360 degrees. Therefore Pi radians is 
   180 degrees and 1/2 Pi radians is 90 degrees. 
 - e raised to the i*pi power equals -1 (e is the base of the natural 
   logarithm and i is the imaginary number which is the sqare root 
   of -1). 
 - Pi day is celebrated on March 14 at the Exploratorium in San 
   Francisco (March 14 is 3/14).
 - All the digits of Pi can never be fully known. 
 
For more about Pi see the Dr. Math FAQ:
  http://mathforum.org/dr.math/faq/faq.pi.html  

The Relationship between Circumference                 Name _________________________________

and Diameter                                                                     Date __________________________________

 

A.     Today we will use different circular objects to discover the relationship between circumference and diameter. First, we need to check to be sure that we understand the difference between these two terms. Use the glossary of your book to write your own definition for the following:

 

Circumference: _______________________________________________________________________

Diameter: _____________________________________________________________________________

 

B.       After you are placed in your groups, you will be given a piece of string and a meter stick. You will go around the room in order and record the diameter and circumference of the circles at the five stations. Listen to the teacher’s directions to know when to change stations. Fill in the first three columns only, until given further directions.

 

Object:

 

Your circumference:

Your diameter:

Class circumference:

Class Diameter:

C x D

C ÷ D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Complete the rest of the chart, following your teacher’s instructions. Then, create a scatterplot that shows the relationship between circumference and diameter.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Use the table and the graph to answer the following questions (on your own paper!)

 

1.  Draw a line of best fit through the scatterplot. Then write an equation to represent the

relationship between circumference and diameter (in the form y=mx+b).

2.  Make a general statement that describes the relationship between diameter and

circumference.

3.  What is the circumference of a circle whose diameter is 9 centimeters? 15 inches? 100

feet? Explain how you found these values.