Which Solids Can We Make?
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 |
| Tetrahedron | ||||
| Octahedron | ||||
| Icosahedron | ||||
| Dodecahedron |
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