cone, sphere, cylinder

a ppt is here

these solids have the same value for the radius and a common 'height'

what is the ratio of volumes of
cone : sphere : cylinder
in its simplest form?

establish that the volume of the sphere plus the cone make the volume of the cylinder

popcorn suitably flattened on top

the result for the relationship between the volumes of a sphere, a cone and a cylinder was allegedly established by Archimedes using small slices

this is explained in 'Cut the Knot' and illustrated here

it can be shown that the volume of the thin disk is the same as the volume of the ring

there's an illustration (with the above diagram) with an explanation of Archimedes' method on a blog by Zachary Abel

there's a youtube video here

cone  +  hemisphere  =  cylinder