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

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

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This proof is in 3D, 2D is much easier, but uses the same logic.
Sorry but this didnt help much. How is this cube related to a circle or a sphere?
I guess the proof for 3D + sphere is similar to the 2D and circle. But how does this cube section help?
It is the starting point: as you increace the number of faces, the volume goes up

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