Construct foam or clay models of primitive solid shapes, such as cylinders, cones, rectilinear prisms, cubes, etc.
a. Intersect the primitive solid with a plane: Cut the primitive into two pieces, and place a (preferably transparent) plane between the two halves. By lifting off one half, you can see the cross section of the intersection, along with the orientation of the cutting plane. Using the cone, create a variety of conic sections by intersecting the plane at different orientations. What is the relationship of the orientation of the intersecting plane to the shape of the intersection? When is the shape and size of the intersection the same as the base? Repeat these tests with the other primitive solids. Also, compare the intersections of different solids with the planes in similar orientations.
b. Intersect two solids with each other: Keep one solid whole and cut the other solid into distinct pieces representing volumes internal and external to the first solid. Create a cone and a rectilinear prism for which one dimension is twice the diameter of the base of the cone and the other two dimensions are two-thirds the diameter. Cut the prism to represent intersecting half-way through the cone at its base. You should end up with the prism in three pieces: two external to the cone and one internal. Cut up three other identical prisms representing the intersections as the prism moves up vertically toward the tip of the cone. Compare the internal and external pieces of the various intersections. When do the internal pieces look more like pieces of the cone? Pieces of the prism? How would the pieces have to change if none of the sides of the prism were parallel to the base of the cone? Try intersecting the prism with the cylinder and then the cube, at both normal and skew angles.