Round Peg fits in Square Hole better than Square Peg fits in Round Hole

Theorem
A round peg fits better in a square hole than a square peg fits in a round hole.


 * SquareAndRoundPegsAndHoles.png

Proof
The situation is modelled by considering the ratios of the areas of:
 * a square to the circle in which it is inscribed
 * a square to the circle around which it has been circumscribed.

Let a square $S$ be inscribed in a circle $C$ of radius $r$.

Let $A_c$ and $A_s$ be the areas of $C$ and $S$ respectively.

From Area of Circle:
 * $A_c = \pi r^2$

The diameter of $S$ is $2 r$.

Thus from Pythagoras's Theorem its side is of length $r \sqrt 2$.

From Area of Square:
 * $A_s = 2 r^2$

Thus:
 * $\dfrac {A_s} {A_c} = \dfrac {2 r^2} {\pi r^2} = \dfrac 2 \pi \approx 0.6366 \ldots$

Let a square $S$ be circumscribed around a circle $C$ of radius $r$.

Let $A_c$ and $A_s$ be the areas of $C$ and $S$ respectively.

From Area of Circle:
 * $A_c = \pi r^2$

The side of $S$ is of length $2 r$.

From Area of Square:
 * $A_s = 4 r^2$

Thus:
 * $\dfrac {A_c} {A_s} = \dfrac {\pi r^2} {4 r^2} = \dfrac \pi 4 \approx 0.7853 \ldots$

Thus a round peg takes up more space ($0.7853 \ldots$) of a square hole than a square peg takes up ($0.6366 \ldots$) of a round hole.