Lesson 2.7 Polyhedra pp. 73-77

Objectives: 1. To define polyhedron and related terms. 2. To identify names of the simple polyhedra. 3. To classify polyhedra as regular or nonregular.

Prisms and pyramids differ from spheres because they have flat faces Prisms and pyramids differ from spheres because they have flat faces. Such closed surfaces are called polyhedra.

Definition A polyhedron is a closed surface made up of polygonal regions. A face of a polyhedron is one of the polygonal regions that form the surface of the polyhedron.

The polyhedra that we will study in the course are simple polyhedra The polyhedra that we will study in the course are simple polyhedra. A polyhedron that is not simple is one that has a hole in it.

Polyhedra are named according to the number of faces. No. of Faces Name 4 tetrahedron 5 pentahedron 6 hexahedron 7 heptahedron 8 octahedron 10 decahedron

Polyhedra are named according to the number of faces. No. of Faces Name 12 dodecahedron 20 icosahedron

EXAMPLE Classify the polyhedra.

EXAMPLE Classify the polyhedra.

EXAMPLE Classify the polyhedra.

EXAMPLE Classify the polyhedra.

EXAMPLE Classify the polyhedra.

Practice: Name the polyhedron.

Practice: Name the polyhedron.

Definition A regular polyhedron is a convex polyhedron having two properties. (1) All faces are identical (Same size and shape), and (2) The same number of edges meet at each vertex.

There are only 5 possible regular polyhedra 1. Regular tetrahedron 2. Regular hexahedron 3. Regular octahedron 4. Regular dodecahedron 5. Regular icosahedron

Regular hexahedron

The intersection of adjacent faces of a polyhedron is called an edge of the polyhedron. The endpoints of the edges are called the vertices.

Homework pp. 76-77

►A. Exercises Tell whether the statement is true or false. 1. Every polyhedron is a cone.

►A. Exercises Tell whether the statement is true or false. 3. Some cones are polyhedra.

►A. Exercises Tell whether the statement is true or false. 5. A prism has only one base. A prism is a cylinder with polygonal regions as bases.

►A. Exercises Tell whether the statement is true or false. 7. The smallest number of vertices of a polyhedron is four.

►A. Exercises Tell whether the statement is true or false. 9. A prism has the same number of faces as vertices.

►A. Exercises Classify the polyhedron according to the number of faces. 13.

►A. Exercises Classify the polyhedron according to the number of faces. 15.

►B. Exercises Give another name for each figure. 21. A prism with a decagonal base region.

►B. Exercises A diagonal of a prism joins two vertices that do not lie in the same face. (This also means that a diagonal must intersect the interior of the prism.) 24. Complete the table Bases of Prism Diagonals/Vertex Total Diagonals quadrilaterals pentagons hexagons octagons

►B. Exercises 24. H G AG BH E F CE DF D C A B

►B. Exercises A diagonal of a prism joins two vertices that do not lie in the same face. (This also means that a diagonal must intersect the interior of the prism.) 24. Complete the table Bases of Prism Diagonals/Vertex Total Diagonals quadrilaterals pentagons hexagons octagons 1 4

►B. Exercises 24. AK L K G J AJ AI H I F E A D B C

►B. Exercises 24. BL L K G J BK BJ H I F E A D B C

►B. Exercises 24. CG L K G J CL CK H I F E A D B C

►B. Exercises A diagonal of a prism joins two vertices that do not lie in the same face. (This also means that a diagonal must intersect the interior of the prism.) 24. Complete the table Bases of Prism Diagonals/Vertex Total Diagonals quadrilaterals pentagons hexagons octagons 1 4 3 18

■ Cumulative Review For solids bounded by the given surfaces, decide whether each is convex or concave. 26. sphere

■ Cumulative Review For solids bounded by the given surfaces, decide whether each is convex or concave. 27. regular polyhedron

■ Cumulative Review For solids bounded by the given surfaces, decide whether each is convex or concave. 28. torus (doughnut-shape)

■ Cumulative Review For solids bounded by the given surfaces, decide whether each is convex or concave. 29. oblique circular cone

■ Cumulative Review For solids bounded by the given surfaces, decide whether each is convex or concave. 30. What geometric figure represents the core of a roll of paper towels? What shape results if you flatten the roll?