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# Thinking in (Semi-)Circles: The Area of the Arbelos

 Difficulty Time Required Short (2-5 days) Prerequisites Must understand the concept and method of a mathematical proof Material Availability Readily available Cost Very Low (under \$20) Safety No issues

## Abstract

The arbelos is the white-shaded region between the three semicircles in the illustration at right. In this project, you'll prove an interesting method for determining the area of the arbelos.

## Objective

Prove that the area of the arbelos (white shaded region) is equal to the area of circle CD.

## Credits

Andrew Olson, Science Buddies
Professor David Joyce, for the Geometry Applet

### MLA Style

Science Buddies Staff. "Thinking in (Semi-)Circles: The Area of the Arbelos" Science Buddies. Science Buddies, 31 May 2013. Web. 28 July 2015 <http://www.sciencebuddies.org/science-fair-projects/project_ideas/Math_p012.shtml>

### APA Style

Science Buddies Staff. (2013, May 31). Thinking in (Semi-)Circles: The Area of the Arbelos. Retrieved July 28, 2015 from http://www.sciencebuddies.org/science-fair-projects/project_ideas/Math_p012.shtml

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Last edit date: 2013-05-31

## Introduction

Important: You will need the current version of Java installed on your computer for this project. If you do not have Java, some figures and images in this project may not display properly in your browser. To check for the current version of Java on your machine, visit Java.

Figure 1 below shows an arbelos. What is an arbelos? The arbelos is the white region in the figure, bounded by three semicircles. The diameters of the three semicircles are all on the same line segment, AB, and each semicircle is tangent to the other two. The arbelos has been studied by mathematicians since ancient times, and was named, apparently, for its resemblance to the shape of a round knife (called an arbelos) used by leatherworkers in ancient times.

Figure 1. The Arbelos.

### 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 an image may display stating "This plugin is disabled" or the diagram may 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.

An interesting property of the arbelos is that its area is equal to the area of the circle with diameter CD (see Figure 2, below). CD is along the line tangent to semicircles AC and BC (CD is thus perpendicular to AB). C is the point of tangency, and D is the point of intersection with semicircle AB. Can you prove that the area of circle CD equals the area of the arbelos?

Figure 2. Prove that the area of the arbelos (white shaded region) is equal to the area of circle CD.

## Terms and Concepts

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

• Right triangles
• Circumscribing a circle about a triangle
• Similar triangles
• Area of a circle
• Mathematical proof

## Bibliography

The Math Forum at Drexel University has some good advice on how to build a mathematical proof:

There are many more examples in their FAQ section:

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. Book IV, Proposition 5, and Book VI, Proposition 8 will be helpful:

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## Materials and Equipment

For the proof, all you will need is:

• Pencil
• Paper
• Compass
• Straightedge

## Experimental Procedure

3. Spend some time thinking about the problem and you should be able to come up with the proof.

## Variations

For a more basic project, see:

If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these links:

## Share your story with Science Buddies!

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## If you like this project, you might enjoy exploring these related careers:

### 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. Read more

### 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! Read more

## News Feed on This Topic

, ,
Note: A computerized matching algorithm suggests the above articles. It's not as smart as you are, and it may occasionally give humorous, ridiculous, or even annoying results! Learn more about the News Feed

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