Section 12-1

1. Section 12-1 Exploring Solids

2. Polyhedron • Three dimensional closed figure formed by joining three or more polygons at their side. • Plural: polyhedra

3. Parts of a Polyhedron

4. Face: Each polygon of the polyhedron • Edge: A line segment along which two faces meet • Vertex: A point where three or more edges meet

5. Regular polyhedron • Has faces that are congruent regular polygons Example:

6. Convex Polyhedron • If any two points on its surface can be connected by a segment that lies entirely inside or outside the polyhedron

7. Concave Polyhedron • If the segment goes outside the polyhedron

8. Cross Section • Intersection of a plane and a solid

9. A plane and a solid’s intersection forms different shapes.

10. Examples of a Plane and a Cube’s Cross Sections Square Trapezoid Triangle

11. Example of a Plane and a Sphere’s Cross Section Circle

12. Euler’s Theorem • The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula:

13. Platonic Solids • Tetrahedron: 4 faces • Cube: 6 faces • Octahedron: 8 faces • Dodecahedron: 12 faces • Icosahedron: 20 faces Five regular polyhedra:

14. Regular Tetrahedron 4 4 6 ____ faces, ____ vertices, ____ edges

15. Cube 6 8 12 ____ faces, ____ vertices, ____ edges

16. Regular Octahedron 8 6 12 ____ faces, ____ vertices, ____ edges

17. 12 30 ____ faces, ____ vertices, ____ edges Regular Dodecahedron 20

18. 20 12 ____ faces, ____ vertices, ____ edges Regular Icosahedron 30