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# Surprising Uses of the Pythagorean Theorem

**davidcary**#139

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.

**niko**#141

the comment might have become a bit confusing in the end.

i guess what im really find interesting is that in the right triangle the self similar areas defined by the two sides add up to the area defined by the hypotenuse, such that one might say that the area is split into the two dimensional components, but the lenghts of the sides obviously dont add up taxicab-like to the lenght of the hypotenuse. yet in the end there is this beautiful relationship of the area to the euclidean norm/the hypotenuse.

well in the end i think that i should probably stay humble and let those patterns be as awesome as they are, instead of speculating all too much. sure its nice to think about those patterns, but i just looked at a pair and it was of such beautiful complex structure, that i just felt like i should spend my time rather on using the patterns to construct something nice. the patterns are there anyway and for the pythagorean theorem i actually have quite a nice intuition already, thanks to your article

**anonymous**#142

Maybe I’m just being slow, but I don’t see much more than just manipulation of equations here! It’s not incredibly profound.

**niko**#143

hi Kalid hope you have time:

so if i see c as a vector that can be split into components (a,b) i can just as well see the area defined by c as a vector that can be split into components of the areas defined by a and b.

so when we split the right triangle into two self-similar triangles, we really split the area into the components of its growth in the two orthogonal dimensions…

what do you think of this? i found this really beautiful but i just wonder how to think of it.

then what we see as the area is really the magnitude of the area vector just as the hypotenuse is just the magnitude of the vector c. and what is magnitude? in fact it is just projection in a different coordinate system…

somehow this is really whirling up quite a bit of my understanding of direction. pretty interesting…

**philjilbert**#144

Hi Khalid, I noticed that you applied the relationship for the well known 5x5 = 4x4 + 3x3 Pythagorean to circular areas. Did you know that 6x6x6 = 5x5x5 + 4x4x4 + 3x3x3 would similarly apply to spherical volumes?

**kalid**#145

Hi Nicolay, great question. It’s figure-out-able after some work but not “obvious” (in that you should see it immediately). Thanks for working through it, it’ll help people!

**nicolay**#146

Okay, I just realised the answer to my question - which in retrospect I should’ve inferred from the title of the post.

For anyone wondering:

d = hypotenuse

s = the length of the sides

So we have:

d^2 = s^2 + s^2

d^2 = 2s^2

1/2 * d^2 = s^2

d/sqrt(2) = s

… I guess it was kinda obvious.

**rod**#147

A three-points geometric model of Pi?

Consider this remarkable Pythagorean triangle that defines both a circle and its square. The center of the circle is located at the point where the perpendicular to the hypotenuse connects with the triangle’s long side; circle’s diameter = 4:

Long side = Pi = 3.14159265358979323846264…

Hypotenuse = (square root of Pi) x 2

= 3.54490770181103205459633…

= side length of circle’s square.

The related circle-squaring inscribed scalene triangle is inherent in this Pythagorean geometry. The scalene’s short side has length equal to one side of a square inscribed in the circle; the longest side forms a 45-degree angle with the hypotenuse of the right triangle.

**nicolay**#148

Hei Kalid!

Another great article, man! One question: How does one intuitively infer that “A regular side is d/sqrt(2)” where d is the diagonal?

**yana**#149

Wow, that blew my mind! I’d always thought the Pythagorean Theorem was kind of dull, but you gave it a whole new twist. And the whole Area=factor*(line segment)^2 was also pretty amazing. I’d never thought of it that way before. Keep up the good work.

**jack**#152

I think the reason the kinetic energy theorem works is this: The force of an object is mass times acceleration, you square the velocity and multiply by 0.5 because acceleration is measured in squares and velocity is the dirivive of acceleration so to balance out the equation you divide by 2.

**shreya**#153

I find difficult 2 find the uses of the Pythagoras theorems extension…can sum 1 plzz help me find it… Plzz…

**rhean**#156

hi Kalid, I’m a new math teacher. Please recommend a good project about Pythagorean theorem for my students to do.