Throwing You Some Curves: Is Red or Blue Longer? *
|Time Required||Very Short (≤ 1 day)|
|Prerequisites||Must understand the concept of a mathematical proof|
|Material Availability||Readily available|
|Cost||Very Low (under $20)|
This a straightforward, but interesting, project in geometry. It is a good first proof to try on your own. You should be able to figure it out by yourself, and you'll gain insight into a basic property of circles.
Figure 1 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. A large semicircle (AE) with smaller semicircles (AB, BC, CD, DE) inscribed in it.
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.
Andrew Olson, Science Buddies
Alexander Bogomolny, for the idea
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Last edit date: 2017-07-28
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:
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MathematicianMathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering. Read more
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