Name the polygon by the number of sides. 1. 6 ANSWER hexagon 2. 10 ANSWER decagon

Name the polygon by the number of sides. 3. 5 ANSWER pentagon 4. What is a regular polygon? ANSWER a polygon with all congruent sides and all congruent angles

Name the polygon by the number of sides. 5. What is the length of the diagonal of a square with side length 6? ANSWER 6 2

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. a. b.

EXAMPLE 1 Identify and name polyhedron c. 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. a.

EXAMPLE 1 Identify and name polyhedron 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. b. The cone has a curved surface, so it is not a polyhedron. c.

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 it has 5 faces, 5 vertices and 8 edges. ANSWER

GUIDED PRACTICE for Example 1 2. It has a curved square, so it is not a polyhedron. ANSWER

GUIDED PRACTICE for Example 1 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 face, 6 vertices, and 9 edges. ANSWER

EXAMPLE 2 Use Euler’s Theorem in a real-world situation House Construction Find the number of edges on the frame of the house. 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. b. SOLUTION a. The cross section is a square. b. The cross section is a rectangle.

EXAMPLE 4 Describe cross sections c. SOLUTION c. The cross section is a trapezoid.

GUIDED PRACTICE for Examples 2, 3, and 4 4. Find the number of faces, vertices, and edges of the regular dodecahedron on page 796. Check your answer using Euler’s Theorem. SOLUTION Counting on the diagram, the dodecahedron has 12 faces, 20 vertices, and 30 edges. Use Euler’s theorem to F + V = E + 2 12 + 20 = 30 + 2 32 = 32 Check Euler’s theorem Substitute This is a true statement so, the solution check

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 6. ANSWER The cross section is a circle

GUIDED PRACTICE for Examples 2, 3, and 4 7. ANSWER The cross section is a hexagon

Daily Homework Quiz Determine whether the solid is a polyhedron. If it is, name it. 1. ANSWER no

Daily Homework Quiz 2. ANSWER yes; pyramid

Daily Homework Quiz 3. ANSWER yes; pentagonal prism

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