Polygons inside a square

Given a square, we can fit various polygons inside it. We often ask: What is the largest (or smallest) possible area or perimeter of a polygon with a certain number of sides that can fit inside the square? The polygon must be entirely contained within the square.

Important properties

• The largest possible area of a polygon inside a square is at most the area of the square.

• For regular polygons (all sides and angles equal), the largest such polygon inside a square is usually the one whose vertices touch the sides of the square.

• The perimeter of a polygon inside a square cannot be more than the perimeter of the square multiplied by the number of times it 'zigzags' across the square, but is always finite.

• For triangles, the largest area is half the area of the square (using the diagonal as the base).