Square vs rectangle Same perimeter, which has shortest diagonal?

Question

My understanding is that a square will always have a shorter diagonal than a rectangle for a given perimeter. E.g. a square of side 3 will have a diagonal of \sqrt{18}, while a 4x3 rectangle has a diagonal of \sqrt{20}.

Is my understanding correct and, if so, what's the rationale behind the rule?

Is my understanding correct and, if so, what's the rationale behind the rule?

VyshakhR1995 · Answer

In your example perimeter o square is 3*4=12
while that of rectangle is 2(4+3)=14

Cez005 · Answer

Thnx, edited. Anyway, you get my point, right?

ccooley · Answer

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...?

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Square vs rectangle Same perimeter, which has shortest diagonal?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards