KS 2-3: What makes a polyhedron regular? Why can the faces of the solids only be equilateral triangles, squares or regular pentagons? Draw nets of the five solids (try using Logo software). What is the relationship between the numbers of vertices, faces and edges of each solid?
KS 4: Explain why there are only five regular polyhedron. What are the angles between adjacent faces for each of the five solids? As an extension, investigate geodesic domes.
The classic text on 3D geometry and model making is Mathematical Models by HM Cundy and AP Rollett (Tarquin pound;9.95).