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As we know, even Archimedes did soon some experimental calculations.
My question were, who calculated first time the exact formulas ($V=\frac{4\pi}{3}r^3$, $A=4\pi r^2$)?
As I know, these formulas require the higher understanding of differential calculus, thus I think it happened after Newton and Leibnitz. But who did them?
This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to history, is that only Euler presented volume formulas in this form in his textbooks after 1737.
The principal step was no doubt made by Archimedes in On Sphere and Cylinder, where he proved rigorously that $V_S:V_C=2:3$, where $V_S$ is the volume of a sphere and $V_C$ is the volume of the circumscribed cylinder (he also gave proportion for the surface area). "Obviously", $V_C=\pi r^2\times2r$, so we get the modern formula, right? Although this is a popular way of projecting modern concepts onto history it sells short the ingenuity of ancient Greeks and the difficulties they managed to overcome, not to mention the work of countless others who brought about our mathematical paradise. Here is one problem: how does one assign a number to a volume, or even to a length for that matter? Today we use real numbers and integration theory, but ancient Greeks had none of that. Their ingenious solution was to make do without both. Geometric magnitudes (volumes, areas, lengths) weren't assigned numbers at all, they were related to other like magnitudes as ratios. These ratios weren't numbers, they could be compared but not added, and only occasionally they could be expressed as ratios of whole numbers, the only numbers proper. This is why Archimedes expressed the "formula" the way he did.
But to get to our modern formula even in the ratio form like $V_S:r^3=4\pi:3$ there is another problem standing in the way. This proportion relates the volume of the sphere not to a cylinder, but to the cube on its radius. Problem is, $4\pi:3$ is not a ratio of whole numbers, unlike $2:3$. Of course, Archimedes didn't know that for sure, but Pythagoreans already got into hot water by assuming it for the side and the diagonal of a square, and later proving otherwise. So Archimedes, like geometers before and after him, did not write it that way, and they did not write $A=\pi r^2$ or $A:r^2=\pi$ for the area of a circle either. Not in equations and not in words. Yet again, Greeks rose to the occasion despite the absence of our modern machinery. The theory of proportion, an ingenious invention of Eudoxus of Cnidus presented in Book V of Euclid's Elements, allowed them to make sense of estimates like $r:s<A:r^2<p:q$ with whole numbers $p,q,r,s$. Archimedes and many of his successors did prove many such estimates without invoking mysterious entities, which remained undefined for almost two millenia hence, and without the modern idea that unbeknownst to them those ratios were "approximating" $\pi$.
If this last step seems trivial to us today let me point out that in 17th century Cavalieri and Roberval were still presenting their volumes and areas as ratios to other simpler volumes and areas, and recall the history of zero, which was not understood or used as a number for centuries after Babylonians and Alexandrian astronomers were using a symbol for it as a placeholder. With $\pi$ there wasn't even a symbol. Only at the end of middle ages some Arabs and Europeans started thinking of irrationals as some kind of numbers, while giving them telling nicknames, like "deafmute numbers". And this was for irrationals like $1+\sqrt{5}$ or $\sqrt[3]{2}$ given by algebraic formulas.
It appears that the first person to contemplate that $\pi$ and $e$ were also "some kind of numbers" in print was James Gregory in The True Squaring of the Circle and of the Hyperbola published in 1667. He was also the first to suggest a possibility that quadrature of the circle was unsolvable with straightedge and compass, although his argument for it was flawed. Even then it took time for the idea to percolate until William Jones in 1706 was bold enough to assign a symbol to the new "number", our modern $\pi$, while still saying "the exact proportion between the diameter and the circumference can never be expressed in numbers". Integration theory was sufficiently developed by then to be comfortable with volumes and areas as numbers as well, so Jones could dispose with the ratios and write $A=\pi r^2$. And when Euler adopted the symbol 30 years later, and made it famous, he could finally write $V_S=\frac43\pi r^3$.
Archimedes calculated the exact formulas (in the way that the ancient Greeks gave formulas) in his book On the Sphere and Cylinder. This was not "experimental": He gave a full geometric proof, rigorous for its time period.
He considered this his greatest work. He asked that a diagram representing his proof be inscribed on his tomb. This was apparently done as at least one visitor to later Syracuse reported seeing the diagram.
Some people think that Archimedes discovered calculus and found the formulas in that way, but hid his discovery. Newton did much the same thing later, using Calculus to discover much about gravity but using geometric proofs when he wrote about gravity. Newton finally revealed his own work in Calculus when forced by Leibniz. Newton realized that Calculus would be controversial. Perhaps Archimedes did as well.
