Sphere is the object that has the minimal area for a fix volume, proof

[andonrangelov]

Can someone give a proof of the statement: “For fix volume a 3-D objects that has minimal surface area is the sphere” ?

 

[mathman]

http://www.peejeshare.com/mathemati...volume-sphere-has-minimum-surface-area-123391

The above may help.

 

[rbj]

http://www.peejeshare.com/mathemati...volume-sphere-has-minimum-surface-area-123391

The above may help.

didn't seem too helpful. maybe it needs some LaTeX formatting

i could tell some of the ? marks were supposed to be $\pi$, but there were some connections i could not make in the derivation.

 

[micromass]

Start here: http://en.wikipedia.org/wiki/Isoperimetric_inequality

 

[CellCoree]

I can provide evidence and reasoning to support the statement that the sphere is the object with the minimal surface area for a fixed volume.

Firstly, we can consider the definition of surface area. Surface area is the measure of the total area that the surface of an object occupies. It is calculated by adding up the areas of all the individual faces or surfaces of the object. Therefore, for a given volume, the object with the smallest surface area will have the least amount of surface area to cover that volume.

Next, we can look at the properties of a sphere. A sphere is a perfectly symmetrical object, meaning that all points on its surface are equidistant from its center. This symmetry allows the sphere to have the most compact and efficient shape, resulting in the smallest surface area for a given volume. This can be seen by comparing a sphere to other 3-D shapes such as a cube, which has a larger surface area for the same volume.

Furthermore, we can use mathematical equations to prove that the sphere has the minimal surface area for a fixed volume. According to the isoperimetric inequality, for any given volume, the sphere has the smallest surface area among all 3-D objects. This can be mathematically proven using the isoperimetric inequality theorem.

In conclusion, the evidence and reasoning presented support the statement that the sphere is the object with the minimal surface area for a fixed volume. The definition of surface area, the properties of a sphere, and mathematical equations all point to the sphere being the most efficient and compact shape for a given volume. Therefore, it can be considered as proof that the sphere is the object with the minimal surface area for a fixed volume.