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 some initial obvious observations, and a quick question:

1. Assuming the square has a side of length 'a', both 0-degree and 45-degree folds (the limits) produce areas of ¨ö a©÷. By observation, these two limits represent the minimum area possible.

2. 0-degree fold produces a perimeter of 3a, while a 45-degree fold produces a perimeter of (2+¡î2)a.

Question: For the single fold not through the center, does it still need to be a nonagon?

Sounds like a job for someone with a spreadsheet, some free time, and a stack of post-it notes.