Square vs rectangle Same perimeter, which has shortest diagonal?

You're correct: a square will always have a shorter diagonal than a rectangle. There are two ways to think about this. One of them is more useful for the GMAT than the other!

The useful way: "Symmetry is a maximizing (or minimizing) principle." If you have two shapes with the same perimeter, the most symmetrical one will have the greatest area, and the smallest diagonal. For example, a square is more symmetrical than a rectangle, and that's where you get the rule you're talking about in your post.

A regular hexagon is even more symmetrical than a square, since it has six identical sides, not four. Correspondingly, suppose you have a square with perimeter 12, and a hexagon with perimeter 12. The square will have an area of 9, and a diagonal of about 4.2. The hexagon will have a larger area, of about 10.4, and a smaller diagonal, of 4.

A circle is even more symmetrical than a hexagon. If you have a circle with perimeter 12, it'll have an even larger area (about 11.5) and an even smaller diagonal (about 3.8).

The more symmetrical a shape is, the bigger area it will fit into a smaller "space". For example, you could enclose more area with the same amount of fence by putting it in a circle, than by putting it in a complex shape, or a square, or a triangle.

The less useful way: You can use algebra to prove that a rectangle has a larger diagonal than a square, specifically. If a square and a rectangle have the same perimeter, then the square will have four sides of length x, x, x, and x. The rectangle will have four sides of length x + ?, x - ?, x + ?, and x - ?. What happens when you calculate the diagonals of those...?

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