Andras 3 years ago Can someone give a proof for this theorem? The circle has the largest area of all shapes, given a fixed perimeter.

1. henpen

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.

2. henpen

|dw:1350744302510:dw|

3. henpen

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

4. henpen

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

5. Andras

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

6. Andras

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

7. henpen

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

8. estudier
9. Andras

thanks