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Throwing You Some Curves: Is Red or Blue Longer?

Summary

Areas of Science
Difficulty
 
Time Required
Very Short (≤ 1 day)
Prerequisites
Must understand the concept of a mathematical proof
Material Availability
Readily available
Cost
Very Low (under $20)
Safety
No issues
Credits

Andrew Olson, Science Buddies
Alexander Bogomolny, for the idea

*Note: For this science project you will need to develop your own experimental procedure. Use the information in the summary tab as a starting place. If you would like to discuss your ideas or need help troubleshooting, use the Ask An Expert forum. Our Experts won't do the work for you, but they will make suggestions and offer guidance if you come to them with specific questions.

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Abstract

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.

The diameter of a red semi-circle is filled with four blue semi-circles of various sizes
Figure 1. A large semicircle (AE) with smaller semicircles (AB, BC, CD, DE) inscribed in it.

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.

Bibliography

The Math Forum at Drexel University has some good advice on How to Build a Proof.

There are many more examples in their FAQ section.

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Careers

If you like this project, you might enjoy exploring these related careers:

Career Profile
Mathematicians 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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General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.

MLA Style

Science Buddies Staff. "Throwing You Some Curves: Is Red or Blue Longer?" Science Buddies, 20 Nov. 2020, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p010/pure-mathematics/red-blue-curves-longer. Accessed 4 Oct. 2022.

APA Style

Science Buddies Staff. (2020, November 20). Throwing You Some Curves: Is Red or Blue Longer? Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p010/pure-mathematics/red-blue-curves-longer


Last edit date: 2020-11-20
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