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

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.*

Solution link:

Peg and hole puzzle: Solution
## Comments

## More!

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

## squares and circles

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

## 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)).

## 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.

## 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

## 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

## 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!

## ...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

## 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

## 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

## 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! )

## What do you mean by tighter

What do you mean by tighter fit?

## 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

## 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