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Vocabulary

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Chapter 11 286

11-1 Space Figures and Cross Sections

Review

Complete each statement with the correct word from the list.

edge edges vertex vertices

1. A(n) 9 is a segment that is formed by the intersections of two faces.

2. A(n) 9 is a point where two or more edges intersect.

3. A cube has eight 9.

4. A cube has twelve 9.

Vocabulary Builder

polyhedron (noun) pahl ih hee drun (plural: polyhedra)

Related Words: face, edge, vertex

Definition: A polyhedron is a space figure, or three-dimensional figure, whose surfaces are polygons.

Origin: The word polyhedron combines the Greek prefix poli-, meaning “many,” and hedron, meaning “base.”

Examples: prism, pyramid

Non-Examples: circle, cylinder, sphere

Use Your Vocabulary

5. Cross out the figure below that is NOT a polyhedron.

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polyhedra

Pyramid Prism

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Problem 2

Key Concept Euler’s Formula

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287 Lesson 11-1

Identifying Vertices, Edges, and Faces

Got It? How many vertices, edges, and faces are in the polyhedron at the right? List them.

6. Identify each description as a vertex, an edge, or a face.

a point where three or polygon a segment where two

more edges intersect or more faces intersect

7. List the vertices.

8. List the edges. Remember to list the dashed hidden edges.

9. List the faces. Remember to list the hidden faces.

10. The polyhedron has vertices, edges, and faces.

Using Euler’s Formula

Got It? Use Euler’s Formula to find the number of faces for a polyhedron with 30 edges and 20 vertices.

11. Use the justifications at the right to find the number of faces.

F 1 V 5 E 1 2 Use Euler’s Formula.

F 1 5 1 2 Substitute with given information.

F 1 5 Simplify.

F 5 2 Subtraction Property of Equality

F 5 Simplify.

12. A polygon with 30 edges and 20 vertices has faces.

The sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number of its edges (E).

F 1 V 5 E 1 2

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Problem 3

Problem 4

Problem 5

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Chapter 11 288

Verifying Euler’s Formula in Two Dimensions

Got It? Use the solid at the right. How can you verify Euler’s Formula F 1 V 5 E 1 2 for the solid?

13. Count the number of vertices.

on the bottom 1 on the top 5 vertices

14. Count the number of faces.

bases 1 lateral faces 5 faces

15. Count the number of edges.

solid edges 1 dashed hidden edges 5 edges

16. Now verify Euler’s Formula for the values you found.

F 1 V 5 E 1 2 Write Euler’s Formula.

1 5 1 2 Substitute.

5 Simplify.

Describing a Cross Section

Got It? For the solid at the right, what is the cross section formed by a horizontal plane?

Underline the correct word to complete each sentence.

17. A horizontal plane is parallel to the bottom / side of the solid.

18. A view from the side / top of the solid helps you see the shape of the cross section.

19. The cross section is a circle / trapezoid .

Drawing a Cross Section

Got It? Draw the cross section formed by a horizontal plane intersecting the left and right faces of the cube. What shape is the cross section?

20. A horizontal plane is parallel to which faces of the cube? Circle your answer.

front and back left and right top and bottom

21. Circle the diagram that shows the intersection of the horizontal plane and the left and right faces of the cube.

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Now Iget it!

Need toreview

0 2 4 6 8 10

Lesson Check

Math Success

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KnowThe octagons and squares are 9.

An edge is a segment formed by the intersection of two 9.

To find the number of 9 without using Euler’s Formula

Count the number of edges formed by the squares and the top octagon.

Count the number of edges formed by the squares and the bottom octagon.

Count the number of edges formed by the squares.

Need Plan

289 Lesson 11-1

Check off the vocabulary words that you understand.

polyhedron face edge vertex cross section

Rate how well you can recognize polyhedra and their parts.

Vocabulary Suppose you build a polyhedron from two octagons and eight squares. Without using Euler’s Formula, how many edges does the solid have? Explain.

24. Complete the problem-solving model below.

25. The intersections of the squares and the top octagon form edges.

26. The intersections of the squares and the bottom octagon form edges.

27. The intersection of the eight squares form edges.

28. The solid has 1 1 5 edges.

• Do you UNDERSTAND?

22. Use the cube to draw and shade the cross section.

23. The cross section is a 9

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