I know this is wrong...

How do you define the radius of a square? You are using half the length of a side but I think you should be using the distance from the center of gravity to the furthest point on the perimeter. So for the square the length of the diagonal is 2r, the length of one side is sqrt(2)*r, the area is 2*r^2, the perimeter is 4*sqrt(2)*r, and the ratio of area to perimeter is r/(2*sqrt(2)) which is smaller than r/2 for the circle.

I'm

of

ea

It doesn't matter what units you use to measure the perimeter or radius as long as they are the same. It will work out to the same ratio. So the que stion is, why is your result different from mine and different still from t he "right" answer of 4/pi between the circle and the square?

Crap! I just saw it. The fact that r remains in the perimeter to area rat io makes the comparison bollocks unless either the area or the perimeter ar e made the same between the circle and the square. That's rather a DUH! W hat does it even mean to take the ratio of area and perimeter?

The analysis that resulted in a ratio of 4/pi between the two was because i t started by equating the perimeters and compared the areas.

So 4/pi it is! I just don't recall if that is an area or perimeter ratio. I guess it must be area.

Ok your algebra checks out. where you fall down is not comparing circles and squares of equal area.

It is well known that larger figures have higher area to perimeter ratios, and above your square is arguably larger than your circle (ask any TV salesman)

BTW, the area of a circle is 0.25*pi*d^2, "not" pi*r^2. It is very hard to measure a radius of a given circle, while measuring its diameter is a no-brainer.

;-P

Best regards, Piotr

The ratios you develop have UNITS of length. That means they aren't pure ratios, they're tied to a measured-value quantity, and that makes the 'r'-dependence a flaw in the attempt to compare ratios. The ratios are like apples and oranges, rather than pure numbers (pi, and 4), and that makes them... incomparable.

Take four separate squares each with side r, perimeter 4r, so the perimeters total 16r.

Now place these four together to make a square with side 2r. But now the perimeter is only 8r - some bastard has nicked half the perimeter!

It's entirely consistent, though. Assume you circle has radius rc, and your square has radius rs, and let the perimeters be the same.

rs = 2 pi rc / 8 = pi rc / 4

Area of square = 4 rs ^2 = 4 (pi rc / 4)^2 = pi^2 rc^2 / 4.

Area of circle = pi rc^2.

Ratio of area of circle to square is 4 / pi.

Sylvia.