![]() This is the method of shell integration in two dimensions. ![]() Using calculus, we can sum the area incrementally, partitioning the disk into thin concentric rings like the layers of an onion. We have indicated where appropriate how each of these proofs can be made totally independent of all trigonometry, but in some cases that requires more sophisticated mathematical ideas than those afforded by elementary calculus. ![]() Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality.Ī variety of arguments have been advanced historically to establish the equation A = π r 2, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can develop trigonometric functions and the fundamental constant π in a way that is totally independent of their relation to geometry. The circumference is 2 π r, and the area of a triangle is half the base times the height, yielding the area π r 2 for the disk. Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle. had found that the area of a disk is proportional to its radius squared. Eudoxus of Cnidus in the fifth century B.C. However, the area of a disk was studied by the Ancient Greeks. Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis. 5.3 Derivation of Archimedes' doubling formulae.
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