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

2. 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

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

4. EXAMPLE 4 Describe the shape formed by the intersection of the plane and the cube. a. b. Describe cross sections

5. EXAMPLE 4 Describe cross sections SOLUTION a. The cross section is a square. b. The cross section is a rectangle. c. The cross section is a trapezoid. c.

6. 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. 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 SOLUTION

7. 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

8. GUIDED PRACTICE for Examples 2, 3, and 4 6. ANSWER The cross section is a circle

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