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EXAMPLE 2

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. House Construction.

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EXAMPLE 2

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  1. 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 House Construction

  2. Use Euler’s Theorem to find the number of edges. F + V = E + 2 Euler’s Theorem 7 + 10 = E + 2 Substitute known values. 15 = E Solve for E. ANSWER The frame of the house has 15 edges. Use Euler’s Theorem in a real-world situation EXAMPLE 2 To find the number of vertices, notice that there are5 vertices around each pentagonal wall, and there are no other vertices. So, the frame of the house has 10 vertices.

  3. 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 Euler’s Theorem 8 + 6 = 12 + 2 Substitute. 14 = 14 This is a true statement. So, the solution checks. Use Euler’s Theorem with Platonic solids EXAMPLE 3

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

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

  6. Check F + V = E + 2 Euler’s theorem 12 + 20 = 30 + 2 Substitute 32 = 32 This is a true statement so, the solution check for Examples 2, 3, and 4 GUIDED PRACTICE 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

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

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

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

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