A Cone, A Sphere, A Cylinder

Explanations

ormulas’ for the volumes of these 3 shapes and respective calculations are as follows. As h=w, all h’s are subsequently replaced with w’s in the process.

Cone: 1/3*pi*r^2*h = 1/3*pi*(w/2)^2*h = 1/12*w^2*h = 1/12*w^3

Sphere: 4/3*pi*r^3 = 4/3*pi*(w/2)^3 = 1/6*w^3

Cylinder: pi*r^2*h = pi*(w/2)^2*h = 1/4*w^3

As it can be seen the three shapes are in ratio of 1/12 to 1/6 to 1/4, or simplified, 1:2:3.

This is one of the amazing fundamental theorems in 3D geometry that the total volume of a cone and a sphere of the same measures equal to the volume of a cylinder of the same measures.