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
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
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Last edit date: 2013-01-10
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.
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 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.
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:
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 - 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'll need is:
- pencil,
- paper,
- compass, and
- straightedge.
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Experimental Procedure
- Do your background research,
- organize your known facts, and
- spend some time thinking about the problem and you should be able to come up with the proof.
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Variations
- For a more basic project, see
Throwing You Some Curves: Is Red or Blue Longer? - If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these links:
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