I know this is wrong...

... but I can't figure out why???

Area to perimeter ratio.

Circle area = pi r^2 perimeter = 2 pi r ratio = r/2

Square area = (2r)^2 = 4 r^2 perimeter = 4 x 2r = 8 r ratio = r/2

But the circle has the highest ratio of area to perimeter! That's a well known fact.

Another proof shows with equal perimeters the areas of a circle and a square are in the ratio of 4/pi.

One of these must be wrong. I'm sure it is the calculations above, but I'm not seeing it. I have to be messing up an assumption.

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  Rick C. 

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Reply to
Ricketty C
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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.

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Regards, 
Carl Ijames
Reply to
Carl

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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.

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  Rick C. 

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Reply to
Ricketty C

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)

--
  Jasen.
Reply to
Jasen Betts

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

Reply to
Piotr Wyderski

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.

Reply to
whit3rd

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!

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Cheers 
Clive
Reply to
Clive Arthur

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

8 rs == 2 pi rc

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.

Reply to
Sylvia Else

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