Name the polygon by the number of sides. 3. 5 1. 6 2. 10 ANSWER hexagon ANSWER decagon ANSWER pentagon 4. What is a regular polygon? ANSWER a polygon with all congruent sides and all congruent angles 5. What is the length of the diagonal of a square with side length 6? ANSWER 6 2

Vocabulary Polyhedron – is a solid bounded by polygons (called faces) that enclose a single region of space. The line segment formed by the intersection of two faces is called an edge. Three or more faces intersect in a single point called a vertex. Bases are two parallel, congruent faces. The solids are named by the shape of their base.

Vocabulary A polyhedron is regular if all of its faces are congruent regular polygons. A polyhedron is convex if the segment joining any two points on the surface lies entirely on the surface and in the interior region. If the polygon is not convex, then it is concave.

Vocabulary Platonic Solids – The five regular polyhedron. 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 http://www.learner.org/interactives/geometry/platonic.html

EXAMPLE 1 Identify and name polyhedra Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. SOLUTION The solid is formed by polygons, so it is a polyhedron. The two bases are congruent rectangles, so it is a rectangular prism. It has 6 faces, 8 vertices, and 12 edges.

EXAMPLE 1 Identify and name polyhedra Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. It has 7 faces, consisting of 1 base, 3 visible triangular faces, and 3 non-visible triangular faces. The polyhedron has 7 faces, 7 vertices, and 12 edges. SOLUTION

EXAMPLE 1 Identify and name polyhedra Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. The cone has a curved surface, so it is not a polyhedron. SOLUTION

GUIDED PRACTICE for Example 1 Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 1. The solid is formed by polygons so it is a polyhedron. The base is a square, so it is a square pyramid. It has 5 faces, 5 vertices and 8 edges. ANSWER

GUIDED PRACTICE for Example 1 Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 2. It has a curved side, so it is not a polyhedron. ANSWER

GUIDED PRACTICE for Example 1 Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 3. The solid is formed by polygons, so it is a polyhedron. The base is a triangle, so it is a triangular prism. It has 5 faces, 6 vertices, and 9 edges. ANSWER

EXAMPLE 2 Use Euler’s Theorem in a real-world situation Find the number of edges on the frame of the house. House Construction SOLUTION 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.

Use Euler’s Theorem in a real-world situation EXAMPLE 2 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 Euler’s Theorem Substitute known values. Solve for E. The frame of the house has 15 edges. ANSWER

Use Euler’s Theorem with Platonic solids EXAMPLE 3 Use Euler’s Theorem with Platonic solids Find the number of faces, vertices, and edges of the regular octahedron. Check your answer using Euler’s Theorem. SOLUTION By counting on the diagram, the octahedron has 8 faces, 6 vertices, and 12 edges. Use Euler’s Theorem to check. F + V = E + 2 8 + 6 = 12 + 2 14 = 14 Euler’s Theorem Substitute. This is a true statement. So, the solution checks.

EXAMPLE 4 Describe cross sections Describe the shape formed by the intersection of the plane and the cube. a. SOLUTION a. The cross section is a square.

EXAMPLE 4 Describe cross sections Describe the shape formed by the intersection of the plane and the cube. b. SOLUTION b. The cross section is a rectangle.

EXAMPLE 4 Describe cross sections Describe the shape formed by the intersection of the plane and the cube. c. SOLUTION c. The cross section is a trapezoid.

GUIDED PRACTICE for Examples 2, 3, and 4 Describe the shape formed by the intersection of the plane and the solid. 5. ANSWER The cross section is a triangle

GUIDED PRACTICE for Examples 2, 3, and 4 Describe the shape formed by the intersection of the plane and the solid. 6. ANSWER The cross section is a circle

GUIDED PRACTICE for Examples 2, 3, and 4 Describe the shape formed by the intersection of the plane and the solid. 7. ANSWER The cross section is a hexagon

Daily Homework Quiz Determine whether the solid is a polyhedron. If it is, name it. 1. 2. 3. ANSWER yes; pentagonal prism ANSWER no ANSWER yes; pyramid

Daily Homework Quiz 4. Find the number of faces, vertices, and edges of each polyhedron in Exercises 1–3. 1. 2. 3. ANSWER pyramid: 5 faces, 5 vertices, 8 edges; prism: 7 faces, 10 vertices, 15 edges

Daily Homework Quiz 5. A plane intersects a cone, but does not intersect the base of the cone. Describe the possible cross sections. ANSWER a point, a circle, an ellipse