Surprising Uses of the Pythagorean Theorem


Ignore that; I’m an idiot. If you treat it abstractly, it’s almost blindingly obvious.

You just have to prove that L^2 / A is constant within a similarity class. Take two members of the same similarity class, of areas A and A’ and lengths L and L’. Let F be the factor of the dilation that maps the first figure onto the other. Then A = F^2 * A’ and L = F * L. Squaring the length equation gives L^2 = F^2 * L’^2. Dividing the area equation by this, the F^2 factors cancel, yielding A / L^2 = A’ / L’^2. So the area to squared length ratio is indeed constant.


Hi Per, glad you wrote it out :). No, you’ve saved me a bit of trouble as I wanted to discuss that aspect too!

... the Pythagorean Theorem depends on the assumptions of Euclidean geometry and doesn't work on spheres or globes...

If a “proof” of the Pythagorean Theorem does not bring in the Euclidean nature of the space under consideration, in what sense can it be considered a proof at all?

According to this wikipedia article, the “Euclidean metric” – which could also be called the “Pythagorean metric” – is actually one of the axioms of a properly-specified Euclidean geometry.

More interesting (to me, at least) would be a discussion of how the Pythagorean Metric leads to our concept of geometric area.

Sorry, I just can’t see how doing it the other way around makes sense.


Hi Ralph, I think discussions of other geometries would be pretty interesting and a good follow-up (I need to read up on them). This wasn’t meant to be a rigorous proof from first principles, more a new intuition onto an old result.


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[…] The Pythagorean Theorem is not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more. […]


Hi Harshil, glad you liked it.


[…] The Pythagorean theorem can apply to any shape, not just triangles. It can measure nearly any type of distance. And yet this 2000-year-old formula is still showing us new tricks. […]


[…] Gelukkig is er BetterExplained, een site vol met betere verklaringen (en dan bedoel ik ook echt beter). In Surprising Uses of the Pythagorean Theorem geeft Khalid Azad een heldere uitleg. Elke rechthoekige driehoek kan in twee kleinere rechthoekige driehoeken verdeeld worden (in het plaatje is gekozen voor a = 3, b = 4 en c = 5, maar het geldt natuurlijk altijd). […]


i need a specific answer why circles are similar


[…] Inside square: side = sqrt(.52 + .52) = .7 [Thanks, Pythagoras], perimeter = 4 * .7 = 2.8 […]


hello, im a school student… and i have a project on pythagorean theorem… and i cant find out in concise the uses/applications of the same in daily life… please help…


[…] We’ve underestimated the Pythagorean theorem all along. It’s not about triangles; it can apply to any shape. It’s not about a, b and c; it applies to any formula with a squared term. […]


how does one calculate the height of i.e. a waterfall using the pathagorean


excellent , mind blowing work, i got a very big help from u for my maths project


@shashank: Glad it was helpful!


I’m doing a research paper and was wondering what your first and last name is so I can site the paper correctly. Thanks!


Hi Ryan, my name is Kalid Azad – good luck with your paper!


this give an complete information on pythagoraus !
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