The X+Y Files

Issue 8

The Pentagon

The United States calls its central Defense (sic) building - which also houses the headquarters of the US Army, Navy and Airforce - the Pentagon. It is a five-story regular pentagon. Pentagons can be constructed, but here is a way to fold one from an A4 sheet of paper.

First, fold the paper corner to corner to create figure 1.

Figure 1

Now fold B onto E to find the axis of symmetry, AM (figure 2).

Figure 2

Open out again and fold BC and ED onto AM. This creates the pentagon APQRS , as in figure 3.

Figure 3

The question is, is it regular? In order to find out we need to show that all the sides and angles are equal. We will discuss this later but here are some ideas that may help.

Try working out all the angles in terms of , =  MAE and ß, =  PQC, when the fold along PQ is opened out again.

You might also try using some trigonometry on triangles APB (in the final position – figure 3) and ABC (in the open position – figure 1).

The sides of the A4 sheet are designed to be in the ratio  
_
1 :

2
 ; in fact they are
about 210 mm and 297 mm. It may be a good idea to work with sides a and ka. If you decide that the pentagon is not regular you may like to decide how much ought to be taken off so that the process does give a regular figure. We are assuming, of course, like the Greeks, that everything is an ideal and not subject to the imperfections of real life.

MLP, from ideas by Nick Lord and Tony Gardiner