Outbound Travel

Why exactly does a cirlcle provide more area than a square each having the same perimeter?

I have pondered on this on many a smoke induced evening. If you take a 10″ piece of string and lay it on a piece of grid paper, you get the most squares inside your string if you make a circle. A square gives you less, and the more you make the square a rectangle, the more area you lose. Eventually, you have zero area left when the walls of your string touch, even though you still have the same outside perimeter. Why is that?

Set up a formula for the sides of a rectangle given the perimeter P, calling one of the sides x. Then the length of the other side is (P-2x)/2 and the area is Px-2x^2/2, or Px/2 – x^2.

You can find the maximum area by differentiating the area with respect to the perimeter. That works out to P/2 – 2x, which you set equal to zero to get the maximum — the point where the area is greatest. If P/2 – 2x =0, then x = P/4.

So you’ve shown that the largest area enclosed by a rectangle comes when the sides are each 1/4 of the perimeter. That’s a square. The same is true for a shape of any number of sides — if the sides are equal, the area is maximum.

Now let’s take a circle with radius r, which means the perimeter is 2 pi(r). Let’s make that the perimeter of a square instead. Each side is 1/4 of that perimeter value, so multiplying the length by the width is squaring pi(r)/2. Work out the algebra and change around the solution to get pi*r^2 times pi/4.

The reason you want the pi*r^2 alone is that that expression is the area of a circle with the perimeter you started with. Now comes the gotcha moment: pi/4 is less than one, so when you multiply the two numbers, the area of the square turns out to be less than the area of the circle.

There’s another way to prove this, but I’m too rusty to remember it. It involves showing that for a given perimeter, the area increases if the shape is changed from a triangle to a square to a regular pentagon to a regular hexagon and so on up. The area is max when you have an infinite number of infinitely small sides on the polygon, which is a circle.

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