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?
Just to be clear (because it certainly wasn't on my computer in my previous post):
Minimum area occurs at 0-degree and 45-degree folds, where the area is 0.5 a^2.
Also, the perimeter at 45-degree fold is (2+sqrt(2))a, and the perimeter at 0-degree is obviously 3a.