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+---- Thread: How do you prove that $\pi<\sqrt{10}$ (/showthread.php?tid=684)
How do you prove that $\pi<\sqrt{10}$ - mklabgr - 06-23-2026
How do you prove that $\pi<\sqrt{10}$
Summary
The article explores a clever mathematical challenge: proving that the famous constant π (pi) is smaller than √10 without simply relying on decimal approximations like π ≈ 3.14 and √10 ≈ 3.16. Instead of using a calculator, the author searches for a pure mathematical argument based on geometry and trigonometry.
The main idea is to use polygons that surround a unit circle. By constructing a regular polygon, its perimeter can be compared with the circumference of the circle, giving an upper bound for π. The author focuses on a 24-sided polygon, because its trigonometric expressions can be simplified into manageable radical forms.
Through a series of algebraic transformations, estimates, and clever use of square roots, the proof shows that the perimeter of this polygon is still smaller than √10. Since the polygon’s perimeter is larger than the circle’s circumference, this leads to the conclusion that π < √10.
What makes the article interesting is not only the result but the problem-solving process: it demonstrates how mathematicians often experiment, test ideas, and work backward from a desired conclusion before creating a clean final proof. It is a great example of how geometry, trigonometry, and creative reasoning come together to prove something that seems obvious but requires careful justification.
ARTICLE