Square Peg, Round PegDate: 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/ |
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