Prove the Pythagoras theorem through calculus

Asked Mar 24 '23 at 09:32

Active Mar 25 '23 at 19:45

Viewed 188 times

Prove the Pythagoras theorem using calculus, by using the fact that the area of a circle is proportional to the square of its radius.

I was reading a routine morning message in a local chat group. There were some interesting facts in the message. One such fact was that we can prove the Pythagoras theorem using calculus, using the fact that the area of a circle is directly proportional to its radius squared.

Now, it intrigued me. I know the proof of Pythagoras theorem using properties of circles (they can be found on this site too) but none of them involved calculus. Even I was not able to apply calculus to prove the famous theorem.

I wonder how we can apply calculus and use the fact that area is proportional to radius squared. Can anyone provide me with an elegant proof$?$

edited Mar 25 '23 at 19:45

Arctic Char

16,007

asked Mar 24 '23 at 09:32

MathStackexchangeIsNotSoBad

4,716

2

This post has proofs by calculus (see the proof by John Molokach). – Dietrich Burde Mar 24 '23 at 10:23
2

IMHO this answer is incorrect. All the construction of derivatives as slope already assumes the euclidean metric in the euclidean plane. – Diego Santos Mar 24 '23 at 21:11
Related: Are there any calculus/complex numbers/etc proofs of the pythagorean theorem? – dxiv Mar 25 '23 at 19:47
Maybe use the fact that $\int_{-a}^a\sqrt{c^2-x^2}dx=\frac{\pi c^2}2$? – Kamal Saleh Mar 25 '23 at 20:00
In my previous comment, the bounds were supposed to be $c$ and $-c$. – Kamal Saleh Mar 25 '23 at 20:20

Prove the Pythagoras theorem through calculus

0 Answers0