1. 12.1 Exploring Solids Hubarth Geometry

2. The three-dimensional shapes on this page are examples of solid figures, or solids. When a solid is formed by polygons, it is called a polyhedron. Polyhedra Prisms and pyramids are examples of polyhedra. To name a prism or Pyramid, use the shape of the base. Rectangular Prism rectangular bases Triangular Pyramid triangular base The two bases of a prism are congruent polygons in parallel planes The base of a pyramid is a polygon

3. Polyhedra Solids cannot have curved surfaces. Cylinder, cone, and sphere shown below are not polyhedra. Cylinder ConeSphere

4. Ex 1 Identify and Name Polyhedra Tell whether the solid is polyhedron. If so, identify the shape of the bases. Then name the solid. a. b. Solution a. The solid is formed by polygons so it is a polyhedron. The figure is a triangular Prism b. A cylinder has curved surfaces, so it is not a polyhedron.

5. Parts of a Polyhedron To avoid confusion, the word side is not used when describing polyhedra. Instead, the plane surfaces are called faces and the segments joining the vertices are called edges. All of the red lines and dotted lines are edges Each corner of the polyhedron is a vertex. The trapezoidal faces of this polyhedron are the bases. So, this is a Trapezoidal Prism

6. Ex 2 Find Faces and Edges Use the diagram at the right. a. Name the polyhedron. b. Count the number of faces and edges. c. List any congruent faces and congruent edges. Solution a. Hexagonal Pyramid b. 7 faces and 12 edges V S R Q P U T

7. Rectangular PrismTriangular Prism Cube Square Prism Rectangular PyramidTriangular PyramidPentagonal Pyramid

8. Theorem 12.1 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. F = 6, V = 8, E = 12 6 + 8 = 12 + 2

9. The frame has one face as its foundation, four that make up its walls, and two that make up its roof, for a total of 7 faces. Find the number of edges on the frame of the house. Ex 3 Use Euler’s Theorem in a Real-World Situation To find the number of vertices, notice that there are 5 vertices around each pentagonal wall, and there are no other vertices. So, the frame of the house has 10 vertices. Use Euler’s Theorem to find the number of edges. F + V = E + 2 7 + 10 = E + 2 15 = E