Regular Polyhedra Answers:

We figured out that there were 5 polyhedra.  You should be able to describe these by name.  For instance you should know:

1. How do you make a tetrahedron? What is it made from? How do you put those pieces together?

It is made from 4 regular triangles that are put together so that 3 triangles meet at each vertex.

2. How do you make a octahedron? What is it made from? How do you put those pieces together?

It is made from 8 regular triangles that are put together so that 4 triangles meet at each vertex.

3. How do you make a icosahedron? What is it made from? How do you put those pieces together?

It is made from 20 regular triangles that are put together so that 5 triangles meet at each vertex.

4. How do you make a cube? What is it made from? How do you put those pieces together?

It is made from 6 squares that are put together so that 3 squares meet at each vertex.

5. How do you make a dodecahedron? What is it made from? How do you put those pieces together?

It is made from 12 regular pentagons that are put together so that 3 pentagons meet at each vertex.

We also figured out how to find the angle measurement of a regular polygon, and we used that to prove that there weren't any more regular polyhedra. You should be able to answer questions like that:

6.A. Find the angle measurement for an angle in an equilateral triangle. Show your work.

For a triangle, you can just remember that the angles in a triangle add up to 180°, and so you can find one angle by dividing by 3: 180° ÷ 3 = 60°

Alternately, you can find the exterior angle by dividing 360° by 3: E = 360° ÷ 3 = 120°.  Then subtract from 180° to get the interior angle A = 180° - 120° = 60°.

B. Explain, using your answer to A, why you can't have a regular polyhedron that has 6 triangles meeting at each vertex.

If you had 6 triangles at a vertex, the total angle would be 6×60° = 360°, which means that it would lie flat and not make a polyhedron.

7. A. Find the angle measurement for an angle in a regular octagon. Show your work.

Find the exterior angle by dividing 360 by 8: E = 360° ÷ 8 = 45°.  Then subtract from 180° to get the interior angle: A = 180° - 45° = 135°.

B. Explain, using your answer to A, why you can't have a regular polyhedron that has regular octagons for faces.

You have to have at least 3 polygons at a vertex to have a polyhedron.  With octagons, 3 polygons would be 3×135° = 405° which is more than 360°.  If you get more than 360°, you can't ever make a polyhedron out of it.