Return to my Mathematics pages
Go to my home page
© Copyright 2003, Jim Loy
Let's
square a pentagon. Squaring an object means to construct a square having the
same area as the object we are squaring. See Geometric Constructions. The famous example
of this is squaring the circle, which was shown to be impossible in 1882, by
Ferdinand Lindemann. But polygons are surprisingly easy to square. So as I
said, let's square a pentagon.
We can divide our pentagon into triangles. In particular, our pentagon is made up of five identical triangles like those in purple in the diagram. There are two simple ways to make a triangle five times as big as another triangle, either use an equal base and five times the altitude, or use five times the base and an equal altitude. Here I chose to made a right triangle with an equal base and five times the altitude (yellow and green triangle in the diagram). This triangle has the same area as our pentagon. We can make a rectangle equal to this triangle by just bisecting the triangle's altitude, giving us the green and blue rectangle.
Now we come to the only clever part of this whole process, squaring a rectangle. In the diagram, we extend DA to E, so that AE = AB. We then bisect DE at F. We draw the circle with center F and radius FE. This circle intersects line AB at G. And AG is one side of our square. We finish drawing the square. You should see how we can do this with any polygon, regular or irregular.
In the circle above, a theorem of Euclid's shows that AG^2 = (AE)(AD), which was the trick we needed. In general, if two chords (AB and CD) of a circle intersect inside the circle at some point P, then (AP)(PB) = (CP)(PD). This is a very useful theorem. In our diagram AG is half of a bisected chord which I didn't finish drawing.
These diagrams were drawn with the programs Cinderella and Paint Shop Pro.