Henry Ernest Dudeney/Puzzles and Curious Problems/188 - Squaring the Circle/Solution 1

Puzzles and Curious Problems by Henry Ernest Dudeney: $188$

The problem of squaring the circle depends on finding the ratio of the diameter to the circumference.

This cannot be found in numbers with exactitude,

but we can get near enough for all practical purposes.

But it is equally impossible, by Euclidean geometry, to draw a straight line equal to the circumference of a given circle.

You can roll a penny carefully on its edge along a straight line on a sheet of paper and get a pretty exact result,

but such a thing as a circular garden-bed cannot be so rolled.

Now, the line below, when straightened out

(it is bent for convenience in presentation),

is very nearly the exact length of the circumference of the accompanying circle.

The horizontal part of the line is half the circumference.

Could you have found it by a simple method, using only pencil, compasses and ruler?

Make a rectangle with one side equal to the diameter and one side $3$ times the diameter.

Then take the diagonal of this rectangle.

This will be quite close to the circumference, as it is $\sqrt {10}$ or approximately $3.162$ times the diameter.