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

by : $188$

 * Squaring the Circle

Solution
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.

Or we can use this construction:


 * Dudeney-Puzzles-and-Curious-Problems-188-solution.png

$AB$ is the diameter of the circle whose center is at $C$.

Bisect the semicircle $AB$ at $D$.

Construct $AE = AC$ and $AF = AC$.

Construct $DE$ and $DF$.

Let $DE$ and $DF$ intersect $AB$ and $G$ and $H$.

Then $DG + GH$ is one quarter the length of the circumference of the circle within a $\dfrac 1 {5000}$ part.