Surprising Uses of the Pythagorean Theorem


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:

#162 - Open in Google Chrome. (Triangles online calculator developed by Jesus S.)

I propose this free online calculator triangles without advertising to help students with geometry, does not perform the duties, because their calculations formulas are not displayed. It is designed in a didactic way to check and view the realized duties.

TrianCal is online calculator triangles that works with any combination of values including sides, heights, angles, the area or perimeter of any triangle, calculating it with the minimum possible value (typically three).

Other functions:

  • Draw the triangle (s) with GeoGebra.
  • Set the range of values in each element.
  • The type of angle.
  • The type of triangle by its angles and sides.
  • Selection of language (English or Spanish).
  • Select the angle type [degrees (°), radians, degrees, minutes and seconds (° ’ ") or degrees and minutes (° ')].
  • Number of decimal places shown in the results (0-15).
  • You can use the arrow keys and the Tab key to navigate through the settings.
  • Drop-down menu to select the values comfortably.
  • Create a link (URL) to the current triangle.
  • An icon mail to communicate with the author.

NOTE: You must use the Google Chrome browser to display correctly TrianCal.

Examples of possible combinations:

  • The area, perimeter and other data (side, height or angle), if the outside equilateral triangle would not need the third data.
  • 2 angles and other data (if the value of the other data is not put aside the value of “a” at the time of drawing the triangle is 10).
  • One side, one high and one angle.
  • 3 heights.
  • 3 sides.
  • 2 heights and perimeter.
  • Any other combination of values.


I’m new on the forum, but I was wondering if you have an intuitive explanation for why the pythagorean theorem works. I’ve looked at a few proofs online, but all of them seem to involve combining or breaking up different shapes, which I don’t find very satisfying.

By the way, thanks a lot for all of the insights that you share. I was about to give up on my hope that all math could be understood in this way when I discovered your website last year during my sophomore year in high school.