Chapter 10

Circles

Graphs are often used to display data. If you look through newspapers and magazines, you will probably see many graphs displaying data, from stock market prices to the cost of pizza! There are many ways to graph data. Some of the most common types of graphs are line graphs, bar graphs, and circle (or "pie") graphs. A bar graph showing the number of hosts on the internet is shown below:

This bar graph and other interesting statistics about the use of the net may be found at this web site:

http://www.nw.com/zone/hosts.gif

The circle graph below was created by The Graphics, Visualization, and Usability Center to display information about the ages of people who use the world wide web:

For further statistics about who uses the web, and how often, click on the link below. The circle graph is from this web site.

http://www.gatech.edu/pitkow/survey/survey-1-1994/graphs/results-general.html

Of course, circles appear in many other ways, all around us. We circles and arcs in wheels and flowers and even in rainbows.

http://www.geom.umn.edu/education/calc-init/rainbow/

Circles were of great interest to the mathematicians of ancient times. Over 3500 years ago, the ancient Egyptians discoverd that if they measured the Circumference of a circle (the distance 'around' the circle) and then divided this measurement by the diameter (the distance 'across; the circle) that they would always get approximately the same number, a little more than 3. From this calculation of the ratio of Circumference to diameter, we can derive the formula for finding the Circumference of a circle if you are given the diameter:. that formula is C=¹d. Some historical information about ¹ is written below:

"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." Towards 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."

This information came from the following website, which has many other interesting facts about the mysterious and intriguing number called ¹. Why is it called ¹? Because ¹ is the first letter in the Greek word for circumference.

http://www.geocities.com/SiliconValley/Pines/5945/his.html

The fascinating number ¹ continues to fascinate modern-day mathematicians, as you will see on the site linked below:

http://www.escape.com/~paulg53/math/pi/index.html

Many ancient and modern mathematicians have devised ways of estimating the value of this fascinating number. The Math Forum website has a feature called "Ask Dr. Math. Anyone send in a math question, and Dr. Math will answer! Try the link below for a question related to ¹:

http://www.forum.swarthmore.edu/dr.math/problems/corinna5.29.96.html

There are many very interesting (and some quite amusing!) pages on the web related to ¹. Some intersecting links are listed below:

http://www.ccsf.caltech.edu/~roy/pi.html

http://www.winternet.com/~mchristi/piday.html

Project

What activities make up your daily life? Collect data on this topic, and design a circle graph. To collect the data, keep a journal for one week of how much time you spend on each of your activities: you might include such categories as sleeping, doing homework, time spent in class, etc. Organize the data in groups with the hours and fractions of an hour. You might need to combine some of the smaller groups into a more general category. You should have at least 8 categories. Now you need to divide the circle into portions for the categories. For example, if you spend 8 hours sleeping, then that is eight twenty-fourths of the day, and would then be eight twenty-fourths of the 360 degrees in a circle. Some example calculations are shown below:

After calculating the sizes of the sectors of the circle for each of your daily activities, use a compass (or Geometry software) to construct a large circle, and draw a circle graph. Measure the angles using a protractor. Label your graph, using the colorful graph (called "Age of Users") which is near the top of this page, as an example.Use colored pens or pencils to color each sector of your circle graph. Do your best work; make this a beautiful and informational circle graph!

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