This little geometric riddle is something that I have thought about way too much, and it's driving me crazy. I know this is incorrect but I can't figure out why:
The diagonal of a square is twice the length of its sides.
Crazy, right? We all know it's not true. The Pythagorean Theorem states that for a triangle with sides a
, and hypotenuse c
a2 + b2 = c2
which means that for triangles formed by a diagonally bisected square with sides of length n
n2 + n2 = c2
where c is the length of the diagonal. Simple algebra gives us
c = √2n
That's what we all learned in geometry, right? The length of the diagonal of a square is √2 times the length of its sides. Physical reality backs up this fact. Grab a ruler and see for yourself. Now then, why does the following "proof" make sense?
Assuming a square with sides of length a
, the distance between opposite corners of the square traveling along the border of the square is 2a
By dividing both sides into two equal segments and alternating directions, we create a new path of the same distance 2a
2(a/2) + 2(a/2) = a + a = 2a
By continually dividing each side into n
equal segments of length a/n
and rearranging them in this manner, we create a path that is arbitrarily close to the diagonal without changing the total length of the path:
n(a/n) + n(a/n) = a + a = 2a
As the number of segments n
approaches infinity, the total length remains unchanged.
lim(n → ∞) 2an/n = lim(n → ∞) 2a = 2a
Thus the length of the diagonal is 2a
or twice the length of a side.
Please, someone tell me why I'm wrong.