Some pictures which might be of some aid. Most are all annoying animated gifs which suck up your cpu so they are not directly on this page. If you have Maple, the included source generates 3d animations that you can rotate and inspect while the animation is going.
Here is the project statement.
This first one suggests that the volume of a pyramid is 1/3 Base times height at least for this cube case. The Maple source
This one illustrates the equal area cross-sections of the cone and pyramid. Maple source
A non-animated version belowMaple source
The figure on right is inside the cylinder but outside the sphere, the figure on left is inside the cone. The Maple source.
The figure below illustrates the how the surface area on the sphere between k0 and k1 and on the cylinder between k0 and k1 are equal. Maple source
Here is a 3D view of these surfaces in a DXF format
which is a format used by CAD programs. There is a free java dxf viewer
which you can
download
It requires java which you can download from Sun
You may need to run the viewer from the commandline. The following
works for me `java -jar dxfview.jar file-to-view.dxf'.
This is a hyperpyramid, a four dimensional pyramid with cubical base. Think of the cube is staying in one place and changing size with time and ``not as something moving'' towards you. Maple source
This is a hypesphere, a four dimensional sphere. Think of the sphere is staying in one place and changing size with time and ``not moving'' back and forth. Maple source