1. Chapter 12 Surface Area and Volume

2. Topics We Will Discuss 3-D Shapes (Solids) Surface Area of solids Volume of Solids

3. Some Vocab We Should Know Equilateral Triangle Polygon Convex Nonconvex Ratio Scale Factor

4. Skill Review

5. Geometry 12.1 Exploring Solids

6. POLYHEDRA A polyhedron is a solid that is bounded by polygons, called faces, that enclose a single region of space. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet.

7. Example of a Polyhedron

8. Let’s Explore some 3-D Shapes For each shape that you receive. State whether it is a polyhedron, if it is state how many faces, edges, and vertices it has.

9. Types of Solids SOLIDPOLYHEDRON?FACESEDGESVERTICES PRISM PYRAMID CYLINDER CONE SPHERE CUBE

10. REGULAR POLYHEDRA A polyhedron is regular if all of its faces are congruent regular polygons.

11. CONVEX POLYHEDRA A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. If this segment goes outside the polyhedron, then the polyhedron is nonconvex, or concave.

12. CONCAVE POLYHEDRA

13. CROSS SECTION Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section.

14. The 5 Regular Polyhedra Platonic Solids Regular Tetrahedron Cube Regular Octahedron Regular Dodecahedron Regular Icosahedron

15. Euler’s Theorem The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F + V = E + 2

16. Example A solid has 10 faces: 4 triangles, 1 square, 4 hexagons, and 1 octagon. How many vertices does the solid have?

17. Example A solid has 11 faces: 5 quadrilaterals and 6 pentagons. How many vertices does the solid have?

18. Example Some quartz crystals are pointed on both ends, and have 14 vertices and 30 edges. If you plan to put a label on one of the faces of a crystal, how many faces do you have to choose from?

19. Example A paper model of a geodesic dome is composed of 180 triangular faces. How many vertices does it have?

20. Example Like a soccer ball, a snub dodecahedron has 12 pentagonal faces. The rest of its 92 faces are triangle. How many vertices does the solid have?

21. Quick Questions What makes a polyhedron a regular polyhedron? How can you find the number of vertices of a polyhedron if you know the number of faces and edges?