Pegs and Holes

"You can't put a square peg in a round hole."

I don't know who originally said that. But, I do know that three separate, reasonably intelligent people have told me that this week. Each treated it as a universal axiom.

However, I offer the following for your consideration.

1. Trigonometry

Let x be an angle measured counterclockwise from the x-axis along an arc of the unit circle.

x

In the case of a right triangle, then the length of the hypotenuse is expressed as:

hypotenuse = opposite / sin (x).

2.  Our goal

Assume that we express the diameter of a circle as:

d = 2r.

In the instant case, our goal is to restrict the hypotenuse of the square to less than the opening provided in the round (presumably circular) hole:

2r > hypotenuse,

or

2r > opposite / sin(x).

Solving for opposite, which is also the dimension of each side of the square:

opposite <= 2r(sin(x)).

Adding the simplifying assumption that in a square, x must be 45deg, we obtain:

opposite < 2r (0.85090352453),

or

opposite < 1.70180704907(r).

3.  Conclusion

Thus it is demonstrated that a square peg may pass through a round hole, provided that each side of the square is less than 1.7 times the radius of the hole. QED.