Well, vector calculus is essentially based on the Pythagorean theorem. The magnitude of a vector is determined this way. This is important wherever one deals with directional phenomena. The Pythagorean theorem is far more than just a theoretical gimmick!
In general, a square is an area, there is just an exact coincidence in the numbers - a square with a side of 3 m + a square with a side of 4 m = a square with a side of 5 m.
There is an exact agreement in the numbers if and only if the squares are arranged in such a way that one of their sides form the two sides a and b and the diagonal c of a rectangle, in other words, three sides of three squares form a right-angled triangle.