Thinking in (Semi-)Circles: The Area of the Arbelos
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Project Summary
| Difficulty | 6 – 7 |
| Time required | Short (several days) |
| Prerequisites | Must understand the concept and method 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 |
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Objective
Objective: Prove that the area of the arbelos (white shaded region) is equal to the area of circle CD.
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.
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, 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:
- 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.
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.
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:
- For more science project ideas in this area of science, see Pure Mathematics Project Ideas
Credits
Andrew Olson, Science Buddies
Professor David Joyce, for the Geometry Applet
Last edit date: 2005-11-21 16:57:30
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