Surprising Uses of the Pythagorean Theorem


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.


Hi, I assume you’ve read the article here?

For me, I also had to get beyond geometric re-arrangement, which might be a coincidence. The argument that helped me was realizing the Pythagorean theorem emerged from similarity, i.e.

  1. Split a shape into two similar parts

  2. The total equals the area of the parts

  3. The area of each part is proportional to length^2

Therefore: the (length of the total)^2 = (length of part a)^2 + (length of part b)^2

There are more algebraic ways to see the theorem too, but that approach works for me geometrically.


Oh wow, that actually makes a lot of sense. I should have read it more carefully, but I mistook it for one of the proofs I’d seen before and didn’t read it carefully.

Sorry about that
: (