Polyhedra, nets and shortest paths
Decide which polyhedra are convex, use Euler's formula, unfold a prism and find the spider's shortest path over a brick.
- Russia: grade 10
Guide
In the Russian programme this is Grade 10 (polyhedra): a polyhedron, convex and non-convex polyhedra, the net. Shortest paths over the surface of a polyhedron and Euler's theorem belong to the advanced level.
A polyhedron is convex if it lies on one side of the plane of each face; then the segment joining any two of its points lies in it. For a convex polyhedron V − E + F = 2 (vertices, edges, faces).
A net is a flat figure of faces that folds up into the surface. For a shortest path over a surface, unfold the faces involved into one plane: the shortest path is a straight segment.
Common slips: thinking every prism is convex, losing the 2 in Euler's formula, and walking along the edges instead of straight over the net.