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

## 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

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.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:

## Materials and Equipment

For the proof, all you will need is:

- Pencil
- Paper
- Compass
- Straightedge

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.## Experimental Procedure

- Do your background research.
- Organize your known facts.
- Spend some time thinking about the problem and you should be able to come up with the proof.

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.## 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:

- Geometry Applet How-to Pages
- Combining Computer Science and Math: Inscribing a Circle in a Triangle Using the Geometry Applet

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.## Ask an Expert

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