We see how it arrived at this interesting discovery. Archimedes imagined a hemisphere and next to her a straight circular cylinder and a straight cone, both basic ones the same to a great circle course of the hemisphere. Something similar to the drawing that we showed to you

Archimedes cut the three figures by a flat parallel to the base of the cylinder and the cone and was wondered how they would be the sections determined by this plane in cylinder, hemisphere and cone.

In the cylinder a circle of radio R is obtained (you do not forget that the radius is half of diameter d). In the sphere it will also be a circle, but its radius will depend on distance d. Looking at the figure following and deciding to you the theorem of Pitágoras, easily you can write that if the radius of the section is r, then r

In the cone the section also will be a circle and now the radius is still more easy to determine looking at the following figure

As the radius of opening of the cone is of 45º, it is that the radius is d. Thus

**Section cylinder = PR ^{2
}= P (r^{2 }+ d^{2}) = Pr^{2 }+ PS^{2 }=Sección hemisphere + Section cone**

The sections as are sliced of the three obtained figures cutting in parallel to the base of the cylinder. It is that, placing the three figures since we have put them and cutting them in fine slices we will have

Slice in cylinder to height d = Sliced in hemisphere + Slice in cone. If for each height d this relation is had, it seems quite clear that

But, as Archimedes knew very well,

Volume cilindro= PR

Volume cono= PR

Volume hemisphere = 2PR

When Cicerón was named treasurer in Sicily (75a. of C.), it discovered, thanks to the inscription that Archimedes had commanded to record, the tomb of this great wise person of the antiquity that their countrymen of Siracusa had lost of view. Cicerón recovered it, but later it became to lose. It does a few years were two tombs that dispute the authenticity…

The sphere can be considered as composed by a pile of vertex pyramids the center of the sphere and base of area very small S on the sphere. This gives an idea than the area of the spherical surface can merit. The volume of the sphere is 4PR

4PR