Among all shapes with the same parameter,which shape has a largest area?

Question

Square,rectangle,trapezum,circle,triangle or any other?

Answer 1

I assume you mean, What shape of a given perimeter has the largest area. The 2-dimensional shape with the largest ratio of area to perimeter is a circle. You can calculate the ratio L/A for any shape, though for complicated shapes it may be tedious. As the shape has more and more sides added to it, it approaches a circle, where the ration 2*pi*r/pi*r**2 will be the maximum. Similarly, in three dimensions, it's a sphere.

In nature, both surfaces and volumes have energies associated with them - surfaces have a positive energy and volumes have a negative energy. Nature will try to minimize the total energy, so things try to move toward a spherical or circular shape (oil films on water, drops of water, etc.). Of course, that's usually difficult for solids, but even the human body will become more spherical if we let it.

Answer 2

A circle and then next would be a shape with 1 less than infinity sides because that would be the closest thing to a circle.

Answer 3

This is Kevin M's secondary account. Mainly putting this so it alerts you and you see the edit I made just now so my efforts aren't for not. synopsis of edit: My first answer that its as close to a line as you can get remains and i proved it showing that if you had a long rectangular prism with sides 10/sqrt(infinity) and length of infinity then you still have a volume of 100 cc but with a SA of 40*sqrt(infinity) cm^2 Also, its not a plane as ^^ said (though you can also get infinite SA) because assuming its square with a thickness equal to 10/sqrt(infinity) (just so its the same as mine) then the sides are Sqrt(100/(10/sqrt(infinity))) = 4root(infinity)*sqrt(10). Making the SA 2*10*sqrt(infinity) which is just half of the line. Note that I'm ignoring the 4 thin sides of the plane as they are infintesimal

Answer 4

it circle - by inspection and intution
however if you find it necessary to prove it out, set each known shape as a ratio of area over perimeter
(pi r^2)/2pi r where 2pi r = parameter for every subsequent shape

Answer 5

circle first

then a regular polygon with "nearly an infinite number" of sides (i.e. very close to a circle)

then regular polygons with increasingly less sides, ending with an hexagon, then pentagon, then regular tetragon (square), then regular triangle.

Answer 6

circle, then square since it's closest to circle in shape.

and it's "perimeter"

Answer 7

circle

Answer 8

circle

Answer 9

circle