Throwing You Some Curves: Is Red or Blue Longer?

Objective

The objective of this project is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Introduction

The figure below shows a semicircle (AE, in red) with a series of smaller semicircles (AB, BC, CD, DE, in blue) constructed inside it. As you can see, the sum of the diameters of the four smaller semicircles is equal to the diameter of the large semicircle. The area of the larger semicircle is clearly greater than the sum of the four smaller semicircles. What about the perimeter?

Your goal is to prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outside semicircle.

Figure 1 (applet or image): Prove that the sum of the perimeters of the inscribed semicircles is equal to the perimeter of the outer semicircle.

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.

The figure truly illustrates the fact that the position of the points along the line is entirely arbitrary: the proof will hold in any case.

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:

radius of a circle,
diameter of a circle,
circumference of a circle,
π,
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
When you are constructing your proof, we sincerely hope that you will not have to resort to the following methods [grin]:
http://www.bluemoon.net/~watson/proof.htm
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: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

Materials and Equipment

For the proof, all you'll need is:
- pencil,
- paper,
- compass, and
- straightedge.
Here's a suggestion for your display: in addition to your background research and your proof, you can make a model of Figure 1 with colored paper. Use a compass and straightedge to construct the semicircles. Cut pieces of string or yarn equal to the arc-lengths of the semicircles. You can use these to demonstrate that the perimeter lengths are indeed equal.

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