why is the radius of a sphere is cube? where's the height?

Answer 1

The radius is the height of a cone whose apex is at the center of the sphere.

Imagine a whole bunch of right regular pyramids with their apicies (top points) at the center of the sphere. All the pyramids together fill the interior of the sphere. Now, puff out their bases a little to form the sphere's surface. (I'm ducking the calculus parts here with cute images.)

The pyramids all have the same height: the radius of the sphere. The "base" area is the sum total of all the pyramid bases. As it turns out, the surface area of the sphere depends on the square of the radius (which makes sense if you learn to think about it in the right way).

Is that enough explanation?

Answer 2

V = ( 4 / 3 ) π r ³
r ³ = ( 3 V ) / ( 4 π )
r = ( 3 V / 4π ) ^(1/3)
There is no height---only radius.