Which Solids Can We Make?

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Sometimes it helps to think about two dimensions to get a better understanding of what happens in three dimensions:
In two dimensions, the interior angles of convex polygons are always less than $180^{\circ}$. Can you explain why?
In three dimensions, the sum of the interior angles at each vertex of convex polyhedra must be less than $360^{\circ}$. Can you explain why?

The exterior angles of polygons are a measure of how far short the angles are from $180^{\circ}$.
The angle deficit at a vertex of a polyhedron is a measure of how far short each angle sum is from $360^{\circ}$.
The sum of the exterior angles of a polygon is always $360^{\circ}$.
What do you notice about the total angle deficit of a solid?
This table might help:
  Angle Sum  Angle Deficit  Number of Vertices  Total Angle Deficit 
Cube  270 90 8 720