## A Practical Demonstration of the Pythagorean Theorem

The Pythagorean Theorem is probably the most popular theorem in school mathematics. Surely, you have heard or read about it at least once from elementary school to high school. The Pythagorean Theorem states that given a right triangle with shorter sides $a$, $b$, and hypotenuse $c$, the following equation holds

$c^2 = a^2 + b^2$.

## Irrational Lengths and The Root Spiral

When calculators were not yet invented, it was hard for mathematicians to approximate irrational numbers. The irrational numbers was the offshoot of the discovery that the side length of the diagonal of a square with side length 1 is irrational. But how do mathematicians of the ancient time approximate a segment with length, say, $\sqrt{5}$? Can they draw a segment whose length is exactly $\sqrt{5}$?

With the knowledge of the Pythagorean theorem, it is possible to create a right triangle with side 1 unit making the diagonal $\sqrt{2}$. The diagonal can then be used as the side of another right triangle whose shorter side length is 1. This process can go on producing the figure above. » Read more

## The Mathematical Reason Why Manholes are Round

Have you ever wondered why manholes are circular?

The answer is simple. Using other shapes, it is possible for manhole covers to fall through the hole. For example, a square cover with side length 1 meter can fall through a square manhole even if the lip (stopper) makes the side length of the manhole  less than that of its cover.

To explain further, suppose a 5-cm lip is placed on each side of the hole, then that leaves a square hole with side length 90 cm.  Using the Pythagorean theorem, that hole has diagonal of more than 1.27 meters, large enough to swallow the cover (see 3rd illustration in the 1st figure) with a burp.

On the other hand, the constant diameter of a circular cover ensures that it  does not fall through the circular hole no matter how roughshod (I hope I used the word correctly) it is moved by vehicles.  » Read more

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