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GeometryGeometry

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Lesson 11-1

(For help, go to Lesson 1-3.)

For each exercise, make a copy of the cubeat the right. Shade the plane that containsthe 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

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GeometryGeometry

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Space Figures and Cross SectionsSpace Figures and Cross Sections

Lesson 11-1

Homework

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FeatureLesson

GeometryGeometry

LessonMain

Space Figures and Cross SectionsSpace Figures and Cross Sections

Lesson 11-1

Notes

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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.

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GeometryGeometry

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How many vertices, edges, and faces of thepolyhedron are there? List them.

There are 10 vertices:

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Lesson 11-1

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, andquadrilaterals: ABGF, BCHG, CDIH, DEJI, and EAFJ

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Additional Examples

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Identifying Vertices, Edges and Faces

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GeometryGeometry

LessonMain

Space Figures and Cross SectionsSpace Figures and Cross Sections

Lesson 11-1

Notes

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Euler is pronounced “Oiler.”

Reading Math

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GeometryGeometry

LessonMain

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.

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Lesson 11-1

Quick Check

Additional Examples

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Using Euler’s Formula

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GeometryGeometry

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Space Figures and Cross SectionsSpace Figures and Cross Sections

Lesson 11-1

Notes

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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.

FeatureLesson

GeometryGeometry

LessonMain

Use the pentagonal prism from Example 1 to verifyEuler’s Formula. Then draw a net for the figure and verifyEuler’s Formula for the two-dimensional figure.

F + V = E + 2 Euler’s Formula

Use the faces F = 7, vertices V = 10, and edges E = 15.

Count the regions: F = 7

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Lesson 11-1

7 + 10 = 15 + 2 Substitute the number of faces and vertices.

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.

Draw a net.

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Additional Examples

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Verifying Euler’s Formula

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GeometryGeometry

LessonMain

Space Figures and Cross SectionsSpace Figures and Cross Sections

Lesson 11-1

Notes

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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.

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GeometryGeometry

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Describe this cross section.

The plane is parallel to the triangular base of the figure, so the cross section is also a triangle.

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Lesson 11-1

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Describing a Cross Section

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GeometryGeometry

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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.

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Lesson 11-1

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Drawing a Cross Section

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GeometryGeometry

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1. Draw a net for the figure.

Use Euler’s Formula to solve.2. A polyhedron with 12 vertices and 30 edges has how many faces? 20

16

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Lesson 11-1

Circle

Check students’ drawings; rectangle.

Sample:

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.

Lesson Quiz

11-1

FeatureLesson

GeometryGeometry

LessonMain

Lesson 11-1

(For help, go to Lesson 1-3.)

For each exercise, make a copy of the cubeat the right. Shade the plane that containsthe 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

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GeometryGeometry

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Solutions

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Lesson 11-1

1. 2.

3. 4.

5. 6.

7.

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