cylinder volume = base x height = (pi * r2)* (2r) = 2 pi r3

and

cone volume = 1/3 x base x height = 1/3 x [pi * (2r)2] * (2r) = 8/3 pi r3

    so Archimedes reasoned that

1 x (cone volume + sphere volume) = 1/2 x (4 cylinder volumes)

sphere volume = 2 cylinder volumes - cone volume

sphere volume = 2 x (2 pi r3) - 8/3 pi r3

sphere volume = 4/3 pi r3

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