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

*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.*

## Comments

## As a generalization, is the

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

## Puzzle

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

## What do you mean by tighter

What do you mean by tighter fit?

## Sutton

So which one wastes less space

## square pegs/ roundholes generalisation

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! )

## How does the ratio of areas make a issue

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

## the hole made by the extra

the hole made by the extra area

## round peg in square hole is

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

## Transcription error?

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

## fit round peg in square hole

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

## ...and triangular hole

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

## Well, duh!

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!

## Round peg in Square hole

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

## but for the round peg to fit

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

## ask a carpenter

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.

## Touché sir

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

## Andy

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)).

## squares and circles

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

## More!

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

## The tighter fit would be the

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.

## Very useful in a rocking chair

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.

## size

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