### Packing 3D Shapes

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

### Immersion

Various solids are lowered into a beaker of water. How does the water level rise in each case?

### Investigating Solids with Face-transitivity

In this article, we look at solids constructed using symmetries of their faces.

# Which Solids Can We Make?

##### Stage: 4 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