Sphere volume according to Archimedes

The famous Greek mathematician Archimedes was the first to derive the formula for the volume of a sphere by comparing the volumes of three bodies through clever physical reasoning.

The derivation of the formula for calculating the volume of a sphere is one of Archimedes' greatest mathematical achievements. At his request - geometry homework help , a section through a sphere, a cone and a cylinder was carved on his tomb. Knowledge of this inscription enabled his tomb to be rediscovered in 75 in a cemetery in Sicily.

Archimedes found the formula for the volume of a sphere by physical considerations. He suspended - math homework solver (probably only in his mind) a sphere of radius r, a cone of radius 2r and height 2r and a cylinder of radius 2r and height 2r all made of the same material as shown in Figure 3 so that they could rotate. By calculating the torques, he was able to show that the arrangement is in equilibrium.

Then, according to the law of levers - do my assignment , the following relationship applies between the masses of the bodies and the distances to the centre of mass:

mball⋅2r+mcone⋅2r=mcylinder⋅r,

thus

2⋅mball+2⋅mcone=mcylinder.

Since the densities of the bodies are the same and the formulae for the cone and cylinder volumes were known, Archimedes was able to derive the formula for the sphere volume from this:

2VSphere=VCylinder-2VCone=π(2r)^2⋅2r-2⋅1/3π(2r)^2⋅2r=8/3π r^3,

thus

VSphere=4/3πr^3

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