Theorem. The volume of a sphere is equal to the product of the area of its surface by one third of its radius.
This gives the formula v = (4/3)πr3. This is one of the greatest discoveries of Archimedes. He also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder. This is easily seen, since the volume of the cylinder is πr2 × 2r, or 2πr3, and (4/3)πr3 is 2/3 of 2πr3. It was because of these discoveries on the sphere and cylinder that Archimedes wished these figures engraved upon his tomb, as has already been stated. The Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack of the city Archimedes was killed by an ignorant soldier. Marcellus carried out the wishes of Archimedes with respect to the figures on his tomb.
The volume of a sphere can also be very elegantly found by means of a proposition known as Cavalieri's Theorem. This asserts that if two solids lie between parallel planes, and are such that the two sections made by any plane parallel to the given planes are equal in area, the solids are themselves equal in volume. Thus, if these solids have the same altitude, a, and if S and S' are equal sections made by a plane parallel to MN, then the solids have the same volume. The proof is simple, since prisms of the same altitude, say a/n, and on the bases S and S' are equivalent, and the sums of n such prisms are the given solids; and as n increases, the sums of the prisms approach the solids as their limits; hence the volumes are equal.
This proposition, which will now be applied to finding the volume of the sphere, was discovered by Bonaventura Cavalieri (1591 or 1598-1647). He was a Jesuit professor in the University of Bologna, and his best known work is his "Geometria Indivisilibus," which he wrote in 1626, at least in part, and published in 1635 (second edition, 1647). By means of the proposition it is also possible to prove several other theorems, as that the volumes of triangular pyramids of equivalent bases and equal altitudes are equal.
To find the volume of a sphere, take the quadrant OPQ, in the square OPRQ. Then if this figure is revolved about OP, OPQ will generate a hemisphere, OPR will generate a cone of volume (1/3)πr3, and OPRQ will generate a cylinder of volume πr3. Hence the figure generated by ORQ will have a volume πr3 - (1/3)πr3, or (2/3)πr3, which we will call x.
Now OA = AB, and OC = AD; also (OC)2 - (OA)2 = (AC)2, so that
(AD)2 - (AB)2 = (AC)2,
and π(AD)2 - π(AB)2 = π(AC)2.
But π(AD)2 - π(AB)2 is the area of the ring generated by BD, a section of x, and π(AC)2 is the corresponding section of the hemisphere. Hence, by Cavalieri's Theorem,
(2/3)πr3 = the volume of the hemisphere.
∴ (4/3)πr3 = the volume of the sphere.