Why a rectangle gets the most area
Imagine building a four-sided shape from two pairs of rigid bars. Keep the lengths of all four sides fixed, but push the top corner sideways. The shape becomes a slanted parallelogram: its base is unchanged, yet its top edge no longer sits directly above the bottom one.
Area depends on perpendicular height—the straight-up distance between the parallel bases—not on the length of the slanted sides. As the shape leans, that height falls, so the same boundary lengths enclose less space.
The geometric tradeoff
If the base has length b, the neighboring side has length s, and the angle between them is θ, the height is s sin θ. The area is therefore bs sin θ. Since sine reaches its maximum value of 1 at 90 degrees, the area is greatest when the sides meet at a right angle: the parallelogram is a rectangle.
At other angles, some of the side length runs sideways instead of contributing to height. That sideways component changes the outline but adds no distance between the bases. The area loss is not caused by shorter sides; it comes from using less of those fixed sides to spread the shape out.
A useful limit
This conclusion holds when the parallelogram’s side lengths stay fixed. It does not mean rectangles are always the largest shapes under every constraint: changing the perimeter, the side lengths, or the allowed shape changes the question. But for a given pair of side lengths, setting them perpendicular captures the greatest possible area.
