Problem: The Pentagon and Square

I wonder how many of my readers, amongst those who have not given any close attention to the elements of geometry, could draw a regular pentagon, or five-sided figure, if they suddenly required to do so. A regular hexagon, or six-sided figure, is easy enough, for everybody knows that all you have to do is to describe a circle and then, taking the radius as the length of one of the sides, mark off the six points round the circumference. But a pentagon is quite another matter.

So, as my puzzle has to do with the cutting up of a regular pentagon, it will perhaps be well if I first show my less experienced readers how this figure is to be correctly drawn. Describe a circle and draw the two lines $H B$ and $D G$, in the diagram, through the center at right angles Now find the point $A,$ midway between $C$ and $B.$1 Next place the point of your compasses at $A$ and with the distance $A D$ describe the arc cutting $H B$ at $E.$ Then place the point of your compasses at $D$ and with the distance $D E$ describe the arc cutting the circumference at $F.$ Now, $D F$ is one of the sides of your pentagon, and you have simply to mark off the other sides round the circle. Quite simple when you know how, but otherwise somewhat of a poser.


Having formed your pentagon, the puzzle is to cut it into the fewest possible pieces that will fit together and form a perfect square

Solutions: 1

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Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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  1. (editor's remark: see how to bisect a segment).