How was the formula for a sphere originated?

Answer 1

The formula of a sphere, in modern terms, originates from calculus. Given that the area of a circle

a(r) = pi*r^2

the definite integration of a(r) from -r to r, the area of the circle rotated around a plane, z, leads to

V(r) = definite integral of -r -> r [pi*y^2]dx

where r^2 = x^2 + y^2, from the distance formula, so r^2 = x^2 - y^2

V(r) = definite integral of -r -> r [pi(x^2-y^2)]dx

V(r) = integral of 0 -> r minus integral of -r -> 0

V(r) = pi(r^3 - (r^3)/3) - pi(-r^3 + (r^3)/3), collect terms

V(r) = (2pi*r^3)/3 + (2pi*r^3)/3

V(r) = (4pi*r^3)/3