Take a square piece of paper oriented with its top horizontal. Fold it along any line that passes through the center and forms an angle of between 0 and 45 degrees with the horizontal.
The outline of the resulting shape is a nonagon.

What angle will maximize the perimeter of this nonagon?

What angle will maximize the area of this nonagon?

Is there any other single fold (not through the center) that can do better for either of these?

In my previous post I said that "the area of the nonagon is the area common to the two folds plus the area of four congruent right triangles". I used this to get the angles for maximum area and perimeter. I was sure that the four triangles were congruent from Sketchpad - they are definitely similar. The adjacent pair are congruent and the separated pair are congruent.

Proved it. 3/15/06

 

Edited on March 15, 2006, 9:41 am

Posted by Bractals on 2006-03-11 22:39:30