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    • Circle Peg trying to fit in Square Hole

      Round peg in square hole or square peg in round hole?

      19 March, 2012
      square hole round peg

      Which gives a tighter fit: a square peg in a round hole or a round peg in a square hole?

      Here is a hint.



      This puzzle was contributed by Colm Mulcahy, Associate Professor of Mathematics at Spelman College in Atlanta, Georgia. Colm's own puzzles have appeared in Math Horizons and in the New York Times. He is a long-time columnist for the Mathematical Association of America and has his own blog. You can follow Colm on Twitter.

      Solution link
      Peg and hole puzzle: Solution
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      Anonymous

      19 March 2012

      Permalink
      Comment

      As a generalization, is the same true for any regular n-gon, as opposed to the square, when you look at the inscribed and circumscribed circles and what is the limiting behaviour?

      AW

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      Anonymous

      20 March 2012

      Permalink
      Comment

      So it looks like the round peg in square hole is tighter since:
      Assume radius =1
      Area of circle = pie(r)^2 = 3.14159
      Are of square = 4x (1*1)=4
      Ratio = 3.14159/4 = .78

      while square peg in round hole is:
      Area of square = 4*(1/2 base*height)=4*(1/2*1*1) = 2
      Area of circle = pie(r)^2 = 3.14159
      ratio = 2/3.14159 = .63

      JV

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      Anonymous

      20 March 2012

      Permalink
      Comment

      What do you mean by tighter fit?

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      Jack

      26 March 2019

      In reply to What do you mean by tighter by Anonymous

      Permalink
      Comment

      So which one wastes less space

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      Anonymous

      29 July 2022

      In reply to What do you mean by tighter by Anonymous

      Permalink
      Comment

      Until the clearances are within .001 inch.

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      Anonymous

      21 March 2012

      Permalink
      Comment

      It gets even more interesting when you generalize this problem to more than two dimensions!
      (i.e. cubes-in-spheres/spheres-in-cubes and their higher dimensional equivalents). Which is the better fit
      changes above 8 dimensions (Roughly speaking as you increase the number of dimensions, more and more of the volume of the hypercube is out near its corners - high dimension cubes are qualitatively more like hedgehogs than building blocks! )

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      gnanasenthil

      27 March 2012

      Permalink
      Comment

      In both the cases the area of contact is going to the same, so how does one be a tighter fit over the other

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      Anonymous

      30 March 2012

      In reply to How does the ratio of areas make a issue by gnanasenthil

      Permalink
      Comment

      the hole made by the extra area

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      Anonymous

      30 March 2012

      Permalink
      Comment

      round peg in square hole is easy
      assume side of square is 2
      A-circle=3.1415926.... (pi*1^2)
      A-square=4 (2^2)
      A.C/A.S=.7354

      square in circle is slightly harder
      diameter circle=2
      A-circle=3.14159...
      A-square=sqrt.2^2=2
      2/3.14...=.63662
      round peg is tighter fit

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      Anonymous

      27 July 2014

      In reply to round peg in square hole is by Anonymous

      Permalink
      Comment

      A.C/A.S=0.7854 (vice 0.7354); doesn't change the conclusion, though.

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      Anonymous

      8 April 2012

      Permalink
      Comment

      A piece of one inch diameter dowel rod one inch long, can be fitted closely into a one inch square hole

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      Anonymous

      11 August 2012

      In reply to fit round peg in square hole by Anonymous

      Permalink
      Comment

      You can do even better and make a shape that fits perfectly in a square hole, round hole and triangular hole. See this clip for the excellent TV show QI

      http://www.youtube.com/watch?v=6fUplOcay7E

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      Anonymous

      9 June 2013

      In reply to fit round peg in square hole by Anonymous

      Permalink
      Comment

      One inch DIAMETER. 1x1 = 1. You're treating radius and diameter as exactly the same entity. a square 1x1, of course a one inch diameter will fit into it. Also, length does not matter!

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      Anonymous

      17 April 2012

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      Comment

      For round peg in square hole-
      P = pi x 5 x 5 = 25pi
      H = 5 x 5= 25
      P : H = pi : 1
      __________________________

      For square peg in round hole-
      P = 5 x 5 = 25
      H = 5 x 5 x pi= 25pi
      P : H = 1 : pi
      ________________________________
      Clearly ratio of P : H is greater in 1st case, so the best fit would be the first one.

      -Ajay Porwal

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      Anonymous

      19 May 2012

      In reply to Round peg in Square hole by Anonymous

      Permalink
      Comment

      but for the round peg to fit in the square hole the radius must be half the side length
      and the square in round hole the side length of the square is sqrt.(2) less than the round hole

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      Anonymous

      3 September 2012

      Permalink
      Comment

      As a carpenter, I know the REAL solution to this problem from a more practical point of view.
      Dress the corners on one end of your square dowel and drive the peg in the hole.
      The edges of the hole will shear off the remaining edges of the square dowel.
      If you're actually doing this somewhere besides a sheet of paper, you will find that
      drilling a round hole is much easier than trying to chisel out a square one.
      As a side note, make sure that your dowel stock is kiln dried and it will stay tight for years to come.

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      Anonymous

      7 April 2016

      In reply to ask a carpenter by Anonymous

      Permalink
      Comment

      I've found this very useful, never thought of drilling a square hole

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      Anonymous

      4 September 2012

      Permalink
      Comment

      I think it should be about the relation between diameter(D) and square side (x).

      (1.) Square Peg in a Round Hole

      To fit in, Diameter (D) = x sqrt (2)

      (2.) Round Peg in a Square Hole
      To fit in, Diameter (D) = x

      So, (2) shall be tighter than (1) since square peg required less diameter hole to fit in (which is D/sqrt(2)).

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      Anonymous

      16 February 2013

      Permalink
      Comment

      Squares within circles within squares give some interesting possibilities. See my http://seekecho.blogspot.co.uk/2011/12/same-area.html

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      Anonymous

      10 June 2013

      Permalink
      Comment

      Now try this for 3 dimensions: cube in sphere or sphere in cube. Solve that and try it in 4D! :)

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      Anonymous

      1 December 2014

      Permalink
      Comment

      The tighter fit would be the square peg in the round hole, because the circle hole is smaller than the square hole, so when you put the square peg into the circle hole it would just fit in, because the circle is round and the square peg is not.

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      Anonymous

      27 November 2015

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      Comment

      To hold permanently a tenon that fits into a mortise in a chair (where there is much flexing) use a peg that goes through both.

      A square piece of wood that is the same size (across flats) as the diameter of the hole is rounded off on one end - one end cylindrical, one end rectangular.

      The round part slips into the round hole. A mallet is used to drive it in the rest of the way.

      It will NEVER loosen up.

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      enchanted wombat

      26 May 2019

      Permalink
      Comment

      I think that it depends on how big the circle and square are. =D

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