Square Peg, Round Peg

Date: 08/22/97 at 03:08:38
From: Tony
Subject: Circle puzzle

Please help!

"Which fits better, a square peg in a round hole, or a round peg in a 
square hole?

Many thanks.

Date: 08/22/97 at 15:14:31
From: Doctor Wilkinson
Subject: Re: Circle puzzle

Cute problem!

Suppose you have a circle of radius r, and you fit a square into it
exactly. (Square peg in round hole).  Then the area of the circle 
is pr(squared). The square has diameter 2r. Call the side of the 
square s. 

Then by the Pythagorean Theorem, 

  s^2 + s^2 = (2r)^2 = 4 r^2

so the area of the square, which is s^2, is just 2r^2. 

If you take the ratio of the area of the circle to the area of the 
square, you therefore get pi/2.

On the other hand, if you fit the circle exactly inside a square 
(round peg in square hole), the area of the square is just the square 
of the diameter of the circle, which is 4 r^2. So the ratio of the 
area of the circle to the area of the square is 4/pi. 

This is bigger than pi/2, so the round peg fits better in the square 
hole than the square peg does in the round hole.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/