Surprising Uses of the Pythagorean Theorem


[…] a mathematical truth like a^2 + b^2 = c^2, were a, b, and c are the sides of a right […]


that was the most fantastic idea i ever came across in maths.


Um… kinda confusing, can’t you just list where it would help in reality and give a few examples instead of trying to make it look complicated. Which it’s actually not… but how does it relate to my life?!


my teacher makes this easyer this just makles my head hert. so much im spelling rong. And i agre with post 98.


The Pythagorean Theorem as modern art, featuring a squared circle: (see Pythagorean Pi design)

Geometers easily comprehend that this new concept of Pi simply complements one ratio (Pi) with another (ASR) … and both ratios include the same mysterious and stimulating essence of irrationality! Such is the nature of squared circles.

How not to square the circle?
Believe that it is impossible.


Hi Mary, the idea is that a yacht is essentially a deformed sphere. Spheres have surface area 4 * pi * r^2.

So we can do

50^2 = 40^2 + 30^2

and multiply both sides by 4 * pi to get

4pi50^2 = 4pi40^2 + 4pi30^2

which says “A ball of radius 50 has the same surface area as a ball of radius 40 and a ball of radius 30”. In our case, we’re using ship hulls, vs. perfect spheres, but the relationship is (theoretically) the same.


man this is awesume. it has helped me a lot in my project


[…] “Pythagorean” Theorem. (Scholars suggest Pythagoras learned it in present day India) […]




[…] “Pythagorean” Theorem. (Scholars suggest Pythagoras learned it in present day India) […]


[…] the Pythagorean Theorem (a2 + b2 = c2) we see how the sides of each triangle are […]


[…] Pythagorean theorem is the single most important math equation you’ll ever encounter.  At it’s most obvious it […]


Just curious on your mention of using perimeter to get area on a square. When you use the formula Area = (1/16)P^2 , there is no way that makes sense. Simply using a 10x10 square, you can figure out this equation doesn’t work out when you wind up with a (100/16) answer instead of 100. What is this?


Hi Justin, great question. In the case of a 10x10 we have Perimeter = 40. So we can plug in P = 40 and get:

Area = (1/16) * (40)^2 = (1/16) * 1600 = 100

Hope that helps!


this website is no help at all!!!


Great work!! And a big thank you for confirming some of my own theories . I pretty much used the same reasoning to prove several mathematical conjectures such as Fermat’s Last Theorem. I know. Wiles has already proved it but certainly not like what Fermat would have. Upon seeing Fermat’s equation I immediately said that’s just the Pythagorean theorem. Its almost exactly like you said except the “factor” is N^n-2. But since you have a,b, and c then N^n-2 is a different factor for each variable. But you can get around that by expressing all the variables in terms of c or the hypoteneuse, i.e., a=csine and b =ccos. Gives a very short sweet margin proof more in keeping with what Fermat actually came up with.

I am convinced that if mathematician would actually sit down and think about some of these problems intuitively rather than trying to impress other mathematicians with “rigor” then far more long standing mathematical problems would be solved.


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It turns out that

$$Area = Factor \cdot (line \hspace{1mm} segment)^2$$

is true even when area is measured by some other shape – when area is measured by something other than tiny squares.


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