Surprising Uses of the Pythagorean Theorem


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


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…


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?


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!


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. :stuck_out_tongue:


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.


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?


Wow, that blew my mind! I’d always thought the Pythagorean Theorem was kind of dull, but you gave it a whole new twist. :slight_smile: 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.


Thanks Yana, glad you enjoyed it!


i dont know what to say its helped me a lot


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.


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


How can you write 5 =4+3 ???


Advantages of using pythagorean theorem are not discussed in the notes above.


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


Thank you so much.


not useful at all


Awesome website, a dose of mathematical sanity, this is the real thing - mathematical meaning before mathematical manipulation (though sometimes we seem to need to learn the manipulation first). (By the way Karan (No. 122), Kalid is not saying “5 = 4 + 3”, but “the area of a circle radius 5 = area of a circle radius 4 + area of a circle radius 3”.)


Two Mango’s of 300 gram and 400 gram equal to One mango of 500 gram? assume the seeds are proportional!!


Brilliant article Kalid, it gives wings to ur imagination!!! i would suggest adding some practical uses of this theorem as i had came to this page looking for some daily uses of the theorem but ended up getting amazed by it… :smiley: