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Project Summary

Difficulty	5  –  6
Time required	Very Short (a day or less)
Prerequisites	Must understand the concept of a mathematical proof
Material Availability	Readily available
Cost	Very Low (under $20)
Safety	No issues

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Sponsor

Sponsored by a generous grant from Motorola

Abstract

Objective

The objective of this project is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Introduction

The diagram below shows a semicircle (AE, in red) with a series of smaller semicircles (AB, BC, CD, DE, in blue) constructed inside it. As you can see, the sum of the diameters of the four smaller semicircles is equal to the diameter of the large semicircle. The area of the larger semicircle is clearly greater than the sum of the four smaller semicircles. What about the perimeter?

Your goal is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Notes on How to Manipulate the Diagram

The diagram is illustrated using the Geometry Applet (by kind permission of the author, see Bibliography). If you have any questions about the applet, send us an email at: scibuddy@sciencebuddies.org. With the help of the applet, you can manipulate the diagram by dragging points.

In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the diagram will still appear, but as a plain, still image.

If you click on a point in the diagram, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. If a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. (Note that diagrams will often use only one or two of the above types of points.)

You can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. If you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration.

You can also lift the diagram off the page into a separate window. When you type u or return the diagram is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger.

The diagram truly illustrates the fact that the position of the points along the line is entirely arbitrary: the proof will hold in any case.

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• radius of a circle,
• diameter of a circle,
• circumference of a circle,
• π,
• mathematical proof.

Bibliography

• The Math Forum at Drexel University has some good advice on how to build a mathematical proof:
  http://mathforum.org/library/drmath/view/54693.html
• There are many more examples in their FAQ section:
  http://mathforum.org/dr.math/faq/faq.proof.html
• When you are constructing your proof, we sincerely hope that you will not have to resort to the following methods [grin]:
  http://www.bluemoon.net/~watson/proof.htm
• The Geometry Applet was kindly provided to Science Buddies by its author, Professor David Joyce, who wrote it for illustrating an amazing online edition of Euclid's Elements: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

Materials and Equipment

• For the proof, all you'll need is:
  • pencil,
  • paper,
  • compass, and
  • straightedge.
• Here's a suggestion for your display: in addition to your background research and your proof, you can make a model of the diagram with colored paper. Use a compass and straightedge to construct the semicircles. Cut pieces of string or yarn equal to the arc-lengths of the semicircles. You can use these to demonstrate that the perimeter lengths are indeed equal.

Experimental Procedure

1. Do your background research,
2. organize your known facts, and
3. spend some time thinking about the problem and you should be able to come up with the proof.

Variations

• For a more advanced project, see
  Thinking in (Semi-)Circles: The Area of the Arbelos
• If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these links:
  • Geometry Applet How-to Pages
  • Combining Computer Science and Math: Inscribing a Circle in a Triangle Using the Geometry Applet

• For more science project ideas in this area of science, see Pure Mathematics Project Ideas.

Credits

Last edit date: 2010-02-17 10:10:22

Career Focus

If you like this project, you might enjoy exploring careers in Pure Mathematics.

	Mathematician Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering.			Statistician Statisticians use the power of math and probability theory to answer questions that affect the lives of millions of people. They tell educators which teaching method works best, tell policy-makers what levels of pesticides are acceptable in fresh fruit, tell doctors which treatment works best, tell builders which type of paint is the most durable. They are employed in virtually every type of industry imaginable, from engineering, manufacturing, and medicine to animal science, food production, transportation, and education. Everybody needs a statistician!
	Math Teacher Math teachers love mathematics and understand it well, but much more than that, they enjoy sharing their enthusiasm for the language of numbers with students. They use a variety of tools and techniques to help students grasp abstract concepts and show them that math describes the world around them. By helping students conquer fears and anxieties about math, teachers can open up many science and technology career possibilities for students. Teachers make a difference that lasts a lifetime!			Economist Every country has resources—people, land, raw materials, capital, and machinery—and economists study how those resources are distributed to create the goods that people buy, and the services people need or want. In their studies, economists monitor economic trends and collect data on things like energy costs, inflation, interest rates, exchange rates, business cycles, taxes, and employment levels. Based on their analysis of this data, they develop forecasts of economic activity so that businesses and governments can better plan for the future.

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