FUN MATH PROBLEM — Circling the Square & Vice-Versa

From time to time I will post interesting math problems.

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit!

I will post the answer to the problems two days later, after people have had time to respond.

To post your response, simply send an email to me @
and make your Subject: Fun Problem.

The problem: Which provides the fuller fit? Putting a circular peg in a square hole, or putting a square peg in a circular hole? To get credit, show all work, and justify your answer by expressing each “fit” as a percent.

A few term-clarifications, to help you do this correctly:

a) By “fit,” I mean the ratio of the smaller shape to the larger shape, expressed as a percent. For
example, if a ratio is 4 to 5, that would represent a “fit” of 80 percent.

b) For the circular peg in the square hole, assume that the diameter of the circle equals the side of the
square. For the square peg in a circular hole, assume that the diameter of the circle equals the diagonal of the square.

c) By “fuller fit,” I mean the larger of the two ratios.

Have fun!

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2 Responses to “FUN MATH PROBLEM — Circling the Square & Vice-Versa”

  1. ZeroSum Ruler says:

    The circle in the square is the better fit at ~78%, opposed to the square in the circle’s ~63%.

    I tried working this problem with x as the circle’s radius, which gave a fit ratio of 2/(pi*x) for the square in circle and a fit ratio of pi/4 for the circle in square.

    A comparison between numbers and variables felt a little “apples and oranges”, so I re-did the problem with radius 1, which led to the 78% and 63% fit ratios.

    Algebra is more my thing than Geometry, so I know there’s the real possibility for mitake in my calculations! Please, don’t hold back! I’d love to know the “right” way to do this!

    • Josh Rappaport says:

      Hi ZeroSum,

      Good work! Yes, your answer is correct. I would just round each off to the hundredths place and get 64% and 79%. But your work is correct. You can see my method of getting the answer on my 4/14 answer post.

      Thanks for participating in the Fun Math Problem!
      —  Josh