What is the relationship between a cylinder and a sphere with the same base area and perpendicular height?

For a sphere, the height is equal to the diameter, or two times the radius.

and because both objects have the same base area, then we can take the radius of the sphere equal to R and the radius of the base of the cylinder also equal to R.

The volume of the cylinder is:

Vc = pi*r^2*h

and here we have r = R and h = 2R

V = pi * (R^3) * 2

the volume of a sphere of radius R is:

Vs = (4/3) * pi*R^3

The quotient between them is:

Vc/Vs = 2 / (4/3) = 6/4 = 3/2.

This means that the volume of the cylinder is 3/2 times the volume of the sphere, always, for any value of R.