I know this is wrong......

[Ricketty C]

.... 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.

--

Rick C.

- Get 1,000 miles of free Supercharging
- Tesla referral code - https://ts.la/richard11209

 

[Carl]

\"Ricketty C\" wrote in message
news:0eac21c3-5bf2-4806-95d5-4431c279df1do@googlegroups.com...

... 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.

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.

--
Regards,
Carl Ijames

 

[Ricketty C]

On Saturday, September 5, 2020 at 8:28:28 PM UTC-4, Carl wrote:

\"Ricketty C\" wrote in message
news:0eac21c3-5bf2-4806-95d5-4431c279df1do@googlegroups.com...

... 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.

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.

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 question is, why is your result different from mine and different still from the \"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 ratio makes the comparison bollocks unless either the area or the perimeter are made the same between the circle and the square. That\'s rather a DUH! What 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 it 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.

--

Rick C.

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+ Tesla referral code - https://ts.la/richard11209

 

[Jasen Betts]

On 2020-09-05, Ricketty C <gnuarm.deletethisbit@gmail.com> wrote:

... 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.

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.

 

[Piotr Wyderski]

Jasen Betts wrote:

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

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

 

[whit3rd]

On Saturday, September 5, 2020 at 4:31:24 PM UTC-7, Ricketty C wrote:

... 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.

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.

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.

--

Rick C.

- Get 1,000 miles of free Supercharging
- Tesla referral code - https://ts.la/richard11209

 

[Clive Arthur]

On 06/09/2020 00:31, Ricketty C wrote:

... 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.

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!

--
Cheers
Clive

 

[Sylvia Else]

On 06-Sep-20 9:31 am, Ricketty C wrote:

... 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.

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.