Space Figures and Cross Sections Lesson 11-1 Check Skills You’ll Need (For help, go to Lesson 1-3.) For each exercise, make a copy of the cube at the right. Shade the plane that contains the indicated points. 1. A, B, and C 2. A, C, and G 3. F, D, and G 4. the midpoints of AD CD, EH, and GH Check Skills You’ll Need 11-1

Space Figures and Cross Sections Lesson 11-1 Homework 11-1

Space Figures and Cross Sections Lesson 11-1 Notes A polyhedron is a three-dimensional figure whose surfaces are polygons. Each polygon is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more edges. 11-1

Space Figures and Cross Sections Lesson 11-1 Additional Examples Identifying Vertices, Edges and Faces How many vertices, edges, and faces of the polyhedron are there? List them. There are 10 vertices: A, B, C, D, E, F, G, H, I, and J. There are 15 edges: AF, BG, CH, DI, EJ, AB, BC, CD, DE, EA, FG, GH, HI, IJ, and JF. There are 7 faces: pentagons: ABCDE and FGHIJ, and quadrilaterals: ABGF, BCHG, CDIH, DEJI, and EAFJ Quick Check 11-1

Space Figures and Cross Sections Lesson 11-1 Notes Euler is pronounced “Oiler.” Reading Math 11-1

Space Figures and Cross Sections Lesson 11-1 Additional Examples Using Euler’s Formula Use Euler’s Formula to find the number of edges on a solid with 6 faces and 8 vertices. F + V = E + 2 Euler’s Formula 6 + 8 = E + 2 Substitute the number of faces and vertices. 12 = E Simplify. A solid with 6 faces and 8 vertices has 12 edges. Quick Check 11-1

Space Figures and Cross Sections Lesson 11-1 Notes A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face. 11-1

Space Figures and Cross Sections Lesson 11-1 Additional Examples Verifying Euler’s Formula Use the pentagonal prism from Example 1 to verify Euler’s Formula. Then draw a net for the figure and verify Euler’s Formula for the two-dimensional figure. Use the faces F = 7, vertices V = 10, and edges E = 15. F + V = E + 2 Euler’s Formula 7 + 10 = 15 + 2 Substitute the number of faces and vertices. Draw a net. Count the regions: F = 7 Count the vertices: V = 18 Count the segments: E = 24 F + V = E + 1 Euler’s Formula in two dimensions 7 + 18 = 24 + 1 Substitute. Quick Check 11-1

Space Figures and Cross Sections Lesson 11-1 Notes A cross section is the intersection of a three-dimensional figure and a plane. The cross section is a triangle. The cross section is a rectangle. 11-1

Space Figures and Cross Sections Lesson 11-1 Additional Examples Describing a Cross Section Describe this cross section. The plane is parallel to the triangular base of the figure, so the cross section is also a triangle. Quick Check 11-1

Space Figures and Cross Sections Lesson 11-1 Additional Examples Drawing a Cross Section Draw and describe a cross section formed by a vertical plane intersecting the top and bottom faces of a cube. If the vertical plane is parallel to opposite faces, the cross section is a square. Sample: If the vertical plane is not parallel to opposite faces, the cross section is a rectangle. Quick Check 11-1

Space Figures and Cross Sections Lesson 11-1 Lesson Quiz 1. Draw a net for the figure. Sample: Use Euler’s Formula to solve. 2. A polyhedron with 12 vertices and 30 edges has how many faces? 20 3. A polyhedron with 2 octagonal faces and 8 rectangular faces has how many vertices? 4. Describe the cross section. 5. Draw and describe a cross section formed by a vertical plane cutting the left and back faces of a cube. 16 Circle Check students’ drawings; rectangle. 11-1

Space Figures and Cross Sections Lesson 11-1 Check Skills You’ll Need (For help, go to Lesson 1-3.) For each exercise, make a copy of the cube at the right. Shade the plane that contains the indicated points. 1. A, B, and C 2. A, B, and G 3. A, C, and G 4. A, D, and G 5. F, D, and G 6. B, D, and G 7. the midpoints of AD CD, EH, and GH Original Check Skills You’ll Need 11-1

Space Figures and Cross Sections Lesson 11-1 Check Skills You’ll Need Solutions 1. 2. 3. 4. 5. 6. 7. Original 11-1