ARCHIMEDES AND THE VOLUME OF THE SPHERE
Many know wise person Archimedes, especially by the handles. The calculation of the volume of the sphere was one of the discoveries that Archimedes considered more of whom he did in his life. It got to demonstrate of a very original way that the volume of the sphere is equal to two thirds of the volume of the circumscribed circular cylinder to her. As much it impressed this to him to the same (perhaps because in that then it was spoken of the perfect bodies) that sent in its tomb this figure in memory of the best one of its ideas was recorded.
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 r2 + d2=R2.
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 = PR2
= P (r2 + d2) = Pr2 + PS2 =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
Volume cylinder = Volume hemisphere + Volume cone
But, as Archimedes knew very well,
Volume cilindro= PR3;
Volume cono= PR3/3 and thus was
Volume hemisphere = 2PR3/3 and Volumen sphere = 4PR3/3.
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 4PR3/3. The one of each pyramid will be RS/3 (because the height of each pyramid is R). Adding all the pyramids and removing to R/3 common factor it is
4PR3/3 = Volume sphere = Extreme volumes pyramids = Area sphere x R/3 and thus
Area sphere = 4PR2
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