Regular polyhedron. Convex polyhedron. Platonic solids. Cross section

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1 12.1 Explore Solids Goal p Identify solids. Your Notes VOCABULARY Polyhedron Face Edge Vertex Base Regular polyhedron Convex polyhedron Notice that the names of four of the Platonic solids end in hedron. Hedron is Greek for side or face. Sometimes a cube is called a regular hexahedron. Platonic solids Cross section Copyright Holt McDougal. All rights reserved. Lesson 12.1 Geometry Notetaking Guide 325

2 12.1 Explore Solids Goal p Identify solids. Your Notes VOCABULARY Polyhedron A polyhedron is a solid that is bounded by polygons that enclose a single region of space. Face The faces of a polyhedron are polygons. Edge An edge of a polyhedron is a line segment formed by the intersection of two faces. Vertex A vertex of a polyhedron is a point where three or more edges meet. Base A base is a polygon that is used to name the polyhedron. Regular polyhedron A regular polyhedron is a polyhedron whose faces are all congruent regular polygons. Convex polyhedron A convex polyhedron is a polyhedron such that any two points on its surface can be connected by a line segment that lies entirely inside or on the polyhedron. Notice that the names of four of the Platonic solids end in hedron. Hedron is Greek for side or face. Sometimes a cube is called a regular hexahedron. Platonic solids A Platonic solid is one of five regular polyhedra: a regular tetrahedron, a cube, a regular octahedron, a regular dodecahedron, and a regular icosahedron. Cross section A cross section is the intersection of a plane and a solid. Copyright Holt McDougal. All rights reserved. Lesson 12.1 Geometry Notetaking Guide 325

3 TYPES OF SOLIDS Polyhedra Not Polyhedra Prism Cylinder Cone Pyramid Sphere Example 1 Describe polyhedra Describe the solid. If it is a polyhedron, find the number of faces, vertices, and edges. a. b. c. Solution a. This is a polyhedron. It has faces so it is a. It has vertices and edges. b. This is a polyhedron. The two bases are congruent hexagons, so it is a. It has faces, vertices, and edges. c. This is not a polyhedron. The solid has a curved surface. 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 1 V 5 E 1. F 5 6, V 5 8, E Lesson 12.1 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

4 TYPES OF SOLIDS Polyhedra Not Polyhedra Prism Cylinder Cone Pyramid Sphere Example 1 Describe polyhedra Describe the solid. If it is a polyhedron, find the number of faces, vertices, and edges. a. b. c. Solution a. This is a polyhedron. It has 4 faces so it is a tetrahedron. It has 4 vertices and 6 edges. b. This is a polyhedron. The two bases are congruent hexagons, so it is a hexagonal prism. It has 8 faces, 12 vertices, and 18 edges. c. This is not a polyhedron. The solid has a curved surface. 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 1 V 5 E 1 2. F 5 6, V 5 8, E Lesson 12.1 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

5 Example 2 Use Euler s Theorem Find the number of faces, vertices, and edges of the polyhedron shown. Check your answers using Euler s Theorem. The polyhedron has faces, vertices, and edges. Use Euler s Theorem to check. F 1 V 5 E 1 2 Euler s Theorem Substitute. 5 Check. Use a triangle to find the height of the triangle in part a. Use a triangle to find the length of a diagonal in part b. Apply the Pythagorean Theorem to find the other lengths you need for parts b and c. Example 3 Describe cross sections Identify and sketch the cross section of the triangular prism. Then find the perimeter and area of the cross section. Every edge of the prism is long. a. b. P 5 A 5 P 5 A 5 c. d. P 5 A 5 P 5 A 5 Copyright Holt McDougal. All rights reserved. Lesson 12.1 Geometry Notetaking Guide 327

6 Example 2 Use Euler s Theorem Find the number of faces, vertices, and edges of the polyhedron shown. Check your answers using Euler s Theorem. The polyhedron has 8 faces, 12 vertices, and 18 edges. Use Euler s Theorem to check. F 1 V 5 E 1 2 Euler s Theorem Substitute Check. Use a triangle to find the height of the triangle in part a. Use a triangle to find the length of a diagonal in part b. Apply the Pythagorean Theorem to find the other lengths you need for parts b and c. Example 3 Describe cross sections Identify and sketch the cross section of the triangular prism. Then find the perimeter and area of the cross section. Every edge of the prism is long. a. b. Isosceles triangle Equilateral triangle 2 3 P 5 12 A 5 4 Ï } P Ï } 2 A 5 4 Ï } 7 2 c. Isosceles d. trapezoid 2 Rectangle P Ï } 5 A 5 3 Ï } P 5 12 A Copyright Holt McDougal. All rights reserved. Lesson 12.1 Geometry Notetaking Guide 327

7 Checkpoint Complete the following exercises. In Exercises 1 3, tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges Is it possible for a polyhedron to have 16 faces, 34 vertices, and 50 edges? Explain. 5. Identify and sketch the cross section of the cylinder. Then find the circumference and area of the cross section. The diameter of the cylinder is 12 and its height is 20. Homework 328 Lesson 12.1 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.

8 Checkpoint Complete the following exercises. In Exercises 1 3, tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 1. This is a polyhedron. Its bases are congruent pentagons, so it is a pentagonal prism. It has 7 faces, 10 vertices, and 15 edges. 2. This is not a polyhedron. The solid has a curved surface. 3. This is a polyhedron. Its base is a rectangle, so it is a rectangular pyramid. It has 5 faces, 5 vertices, and 8 edges. 4. Is it possible for a polyhedron to have 16 faces, 34 vertices, and 50 edges? Explain. No; Using Euler s Theorem F 1 V 5 E 1 2, Þ Identify and sketch the cross section of the cylinder. Then find the circumference and area of the cross section. The diameter of the cylinder is 12 and its height is 20. Circle 6 Homework C 5 12π A 5 36π Lesson 12.1 Geometry Notetaking Guide Copyright Holt McDougal. All rights reserved.