Can someone give a proof for this theorem?
The circle has the largest area of all shapes, given a fixed perimeter.

Question

Can someone give a proof for this theorem?
The circle has the largest area of all shapes, given a fixed perimeter.

anonymous · Answer

Not a formal proof, but here's the intuition.

Imagine a cube, with a centre point. From this point radiate lines that touch the vertices of the cube. Sections such as these|dw:1350744197439:dw| are formed. The area of any pyramid is 1/3 * base area * height. As you add sides to the polyhedron, the height increases, so the volume increases.

anonymous · Answer

|dw:1350744302510:dw|

anonymous · Answer

The total base area for the whole shape is held constant by the requirements of your question,

anonymous · Answer

This proof is in 3D, 2D is much easier, but uses the same logic.

anonymous · Answer

Sorry but this didnt help much. How is this cube related to a circle or a sphere?

anonymous · Answer

I guess the proof for 3D + sphere is similar to the 2D and circle. But how does this cube section help?

anonymous · Answer

It is the starting point: as you increace the number of faces, the volume goes up

anonymous · Answer

http://www.math.utah.edu/~treiberg/isoperim/isop.pdf

anonymous · Answer

thanks