The relationship between parabolas, hyperbolas, circles and ellipses is easy to see once they are understood as conic sections.

I think the relationship between geometric shapes such as squares, rhomboids and trapezoids could be taught in a similar way to conic sections as cross sections of polyhedra
(e.g. a pyramid).

All types of triangles (right angle, equilateral, isosceles, scalene) could be also be visualised as cross-sections of a single polyhedra (i.e. a triangular prism).

This could be used in educational videos or software.

Hmmm. I'm not entirely sure this would
be easier - it depends on which features of
the plane shapes you are trying to
illustrate. Can you give a specific
example?

Hmmm. You mean "slice this
dodecahedron this way and you get a
pentagon" type of thing? Fair enough, but
then you still have to learn about the
properties of the plane shape, once you've
found where it comes from.

I'll agree to it only as a supplement to the current system. There's no harm in offering the same information spun different ways since children have so many different learning styles.