area discrepency on pi*r^2 ?

[AngleWyrm]

Take an ordinary piece of paper and roll it into a cylender. Then crease one
end in four places, making the cylender square on one end, and round on the
other end. The perimiter of the circle end and the square end are exactly
the same length.

Area of the square:
= (perimeter/4)^2
= perimeter^2 / 16

For the circle:
diameter = perimeter / Pi
radius = 0.5 * perimeter / Pi

Pi * radius^2
= Pi * (0.5 * perimeter / Pi)^2
= ( 0.25 perimeter^2 ) / Pi

perimeter^2 / 16 = (0.25 perimeter^2) / Pi only if perimeter is zero.

So does this mean the area covered by the circle is larger than the area
covered by the square?

[Martin Hogbin]

Post by AngleWyrm
Take an ordinary piece of paper and roll it into a cylender. Then crease one
end in four places, making the cylender square on one end, and round on the
other end. The perimiter of the circle end and the square end are exactly
the same length.
= (perimeter/4)^2
= perimeter^2 / 16
diameter = perimeter / Pi
radius = 0.5 * perimeter / Pi
Pi * radius^2
= Pi * (0.5 * perimeter / Pi)^2
= ( 0.25 perimeter^2 ) / Pi
perimeter^2 / 16 = (0.25 perimeter^2) / Pi only if perimeter is zero.
So does this mean the area covered by the circle is larger than the area
covered by the square?

Yes. For a given perimeter, a circle has the largest possible area.
If you crease the cylinder to get a (non-square) rectangle, it will
have less area than the square.


--
Martin Hogbin