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.

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.

$A_c = \pi r^2$

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

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

$A_s = 2 r^2$

Thus:

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

$\Box$

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.

$A_c = \pi r^2$

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

$A_s = 4 r^2$

Thus:

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

$\Box$

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.

$\blacksquare$

1964: David Singmaster: On Round Pegs in Square Holes and Square Pegs in Round Holes (Math. Mag. Vol. 37: pp. 335 – 337) www.jstor.org/stable/2689251