6. A. An icosahedron has 20 faces. Each face is a triangle. Show how to use this information to find the number of edges an icosahedron has.

3 edges per triangle; two faces meet at each edge:20x3/2=60/2=30 edges

B. At each vertex of an icosahedron, 5 triangles meet. Use this and the information in A to find the number of vertices an icosahedron has.

20 triangles; 3 vertices per triangle; 5 triangles per vertex: 20x3/5=60/5=12 vertices

7. To get the number of vertices an icosahedron has, you multiply 20*3 to get the number of vertices on 20 triangles. Then, when you put them together, 5 vertices meet to make 1 vertex, so you divide by 5.

8. A. An octahedron has 8 faces. Each face is a triangle. Show how to use this information to find the number of edges an icosahedron has.

3 edges per triangle; two faces meet at each edge:8x3/2=24/2=12 edges

B. At each vertex of an octahedron, 4 triangles meet. Use this and the information in A to find the number of vertices an octahedron has.

8 triangles; 3 vertices per triangle; 4 triangles per vertex: 8x3/4=24/4=6 vertices

9. A. A tetrahedron has 4 faces. Each face is a triangle. Show how to use this information to find the number of edges a tetrahedron has.

3 edges per triangle; two faces meet at each edge:4x3/2=12/2=6 edges

B. At each vertex of a tetrahedron, 3 triangles meet. Use this and the information in A to find the number of vertices an icosahedron has.

4 triangles; 3 vertices per triangle; 3 triangles per vertex: 4x3/3=12/3=4 vertices

10. A. A cube has 6 faces. Each face is a square. Show how to use this information to find the number of edges a cube has.

4 edges per square; two faces meet at each edge:6x4/2=24/2=12 edges

B. At each vertex of a cube, 3 squares meet. Use this and the information in A to find the number of vertices a cube has

6 squares; 4 vertices per square; 3 squares per vertex: 6x4/3=24/3=8 vertices

11. Tell what the dual of the cube is, and explain how you would create the dual if you had a cube.

The dual of the cube is the octahedron. You could create this by putting a dot in the center of each of the faces of the cube. This would be 6 vertices for the new polyhedron. Then you would connect those dots with edges (an edge would connect the two dots on adjacent faces), and faces.

12. Tell what the dual of the tetrahedron is, and explain how you would create the dual if you had a tetrahedron.

The dual of the tetrahedron is another tetrahedron. You could create this by putting a dot in the center of each of the faces of the tetrahedron. This would be 4 vertices for the new polyhedron. Then you would connect those dots with edges (an edge would connect the two dots on adjacent faces), and faces.

13. Tell what the dual of the icosahedron is, and explain how you would create the dual if you had an icosahedron.

The dual of the icosahedron is the dodecahedron. You could create this by putting a dot in the center of each of the faces of the icoashedron. This would be 20 vertices for the new polyhedron. Then you would connect those dots with edges (an edge would connect the two dots on adjacent faces), and faces.

14. Tell what the dual of the dodecahedron is, and explain how you would create the dual if you had a dodecahedron.

The dual of the dodecahedron is the icosahedron. You could create this by putting a dot in the center of each of the faces of the dodecahedron. This would be 12 vertices for the new polyhedron. Then you would connect those dots with edges (an edge would connect the two dots on adjacent faces), and faces.