For each shape, the constraint is that the shape is inscribed around (or related to) a circle of radius 1.

A square with a distance of 1 from center to side has a side length of $2(1) = 2$, giving an area of:

$A_{\text{square}} = 2^2 = 4$

This is larger than the circle's area ($\pi \approx 3.14$), the regular octagon, the equilateral triangle, and the regular pentagon under the same constraint.

The square has the greatest area.