Challenge Level

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 |