Square Pegs
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Age
11 to 14
Challenge level
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
In other words, which peg fills a greater proportion of its hole?
Image
For the square peg: if the diameter of the hole is $2r$ then the side of the square is $r\sqrt{2}$. The fraction of the area of the cross section of the hole taken up by the peg is $(r\sqrt{2})^2/\pi{r^2} = 2/\pi$.
For the round peg: if the diameter of the peg is $2r$ then the side of the square is $2r$. The fraction of the area of the cross section of the hole taken up by the peg is $\frac{\pi{r^2}}{4r^2} = \frac{\pi}{4}$.
We know that $\pi^2 > 8$ so it follows that the round peg is a better fit as it takes up more of the hole because $\frac{\pi}{4} > \frac{2}{\pi}$.