Asian man solves pi (CNN)

Just kidding LOL.

Is it possible to have two polygons with the same area, but different perimeters?

Is it possible to have two polygons with the same area, and the same (thought differently shaped) perimeters?

Yes.  Consider a right triangle with sides 1, 1, sqrt(2).  It has area 1/2 and perimeter 2+sqrt(2) ~=~ 3.414.

Now consider a rhombus with sides of length (2+sqrt(2))/4.  Such a rhombus can have any area from 0 up to ((2+sqrt(2))/4)^2, depending on the angles.  In particular, there is one choice of angles that gives it area 1/2.

There are presumably many other examples.

Is it possible to squared circles?

>>1
Yes, consider:
#### = 4

##
## = 4

From a practical standpoint, you may have heard that a sphere encloses a volume with minimal surface area.  That implies that it's possible to have a volume enclosed with larger surface areas.  That implies that in 2D, we can have a minimum area enclosure (i.e. a circle), which then implies that there are other examples of larger perimeter.

There was a math problem/proposal that I once read, which challenged the reader to give the minimum area needed to turn a line segment around 180 degrees.  The first answer that comes to mind would be a circle of the segment's diameter.  Closer examination (based upon the 0-width of the line segment) proves that the minimum area is 0 ... since you can move the line segment back and forth with an infinitesimal swing of arc.  The resulting "polygon" looks like a star with infinite spikes of 0 width.  As you can imagine, the perimeter is infinite, too.  Fascinating stuff, although a bit mentally masturbatory.

Anyway, you don't need to be some Asian asshole to solve pi:

π = C/d

>>2
Yes
### - A = 3, P = 8

#
##- A = 3, P = 8

>>4
If you have access to more than just a compass and straightedge, yes.

>>6
Interesting.

>>6 I don't know what you mean.
It seems like the area would be bigger if you're moving the line back and forth :shrug:

>>9
Remember that a line has no width, and therefore no area.

>>6

can you prove a circle encloses the most area for least perimeter, and that a sphere does the same for volume with surface area?

>>11
I think a proof exists, but I can't remember what it is.

>>9
As >>10 said, the line has zero width, which is how you get away with such a thing.  However, study the shape formed by the cam on a Wankel engine.  {googles}  It's called a:

http://en.wikipedia.org/wiki/Reuleaux_triangle

The Reuleaux triangle allows a line to "spin" 360 degrees inside a shape that is about 10% less in area than a circle that uses the same line as the diameter.  The key is that the circle uses a single center, whereas the RT only preserves the line as a "constant width" to the curve that forms its polygon shape, but the center point of the rotation MOVES.

Imagine if you will having an ocean liner that tries to turn around in a bay that is too small to allow a simple central rotation of the vessel.  By using a back-and-forth motion to put the bow and then stern of the vessel into smaller bays of the main bay, the liner is able to eventually turn around in less area.