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
Introduction
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.
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 figure 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 figure will still appear, but as a plain, still image.
If you click on a point in the figure, 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 figures 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 figure off the page into a separate window. When you type u or return the figure 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?
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:
Bibliography
Materials and Equipment
For the proof, all you'll need is:
Experimental Procedure
Variations
Credits
Andrew Olson, Science Buddies
Professor David Joyce, for the Geometry Applet
Last edit date: 2005-11-21 16:57:30
If you like this project, you might enjoy exploring careers in Pure Mathematics.
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