how do we say that among all solids sphere has the least surface area?

Question

please prove it mathematically.

Answer 1

It's not that simple.

Answer 2

The sphere for the same volume as a cube uses less surface area then the cube to contain the same volume.
Proof is as follows;
Let D =one per unit dimension of the diameter of a sphere
Let S = the side of a cube.

Volume of cube =S^3,surface area of cube =6S^2

Volume of Sphere = pi/6 x D^3 = S^3=volume of cube
Solving for area of sphere=pi/6 x D^2= 52.35%D=.5235
Solving for S=80.6% of D
The area of the cube= 6 S^2 = 6(.806D)^2=3.89

area of sphere divided by area of cube =5235/3.89=.1346

The area of the sphere is 13.46 % of the area of the cube.
Therefore; The sphere area is less than the cube area.
That means if you were to contain the same volume of wine in a bottle the Spherical bottle would use less glass surface then the bottle in the shape of the Cube.

Conclusion; The sphere is the most efficient shape to contain the Space the Universe

Answer 3

No matter how you distort a sphere the surface area will always stay the same.
If you force it into a torus,a cube or a pyramid it will maintain it's surface area

Answer 4

Show me a sphere of any size, and I'll show you a cube with smaller surface area.

Do you mean, least surface area for a given volume? That's a different question.