chapter wrap-up chapter summary and 9 vocabulary

Chapter Summary and

Vocabulary9

Chapter Wrap-Up

Summary and Vocabulary 585

The purpose of this chapter is to familiarize you with the common 3-dimensional fi gures, the fi gures of solid geometry. To accomplish this, you should know their defi nitions and how they are related, be able to sketch them, identify plane sections, draw views from different positions, and be able to make some of them from 2-dimensional nets. Below is a hierarchy relating many of these surfaces.

cone pyramid

box

cylindrical surface

surfaces

cylinderprism

sphere

cube

polyhedronconic surface

regularpyramid

regularpolyhedron

right circularcone

While there are many different kinds of 3-dimensional shapes, we focus on the ones in the hierarchy. This is because they tend to be less complicated than irregular shapes and therefore more accessible to study. It is also useful to note that many irregular fi gures often are built from regular ones, or from fi gures that can be approximated by symmetric shapes. Earth can be approximated by a sphere, your neck can be approximated by a cylinder and a volcano can be approximated by a cone.

Many ideas from two dimensions extend to three. The basic properties of planes are given in the Extended Point-Line-Plane Postulate. Like lines, planes may be perpendicular or parallel to each other, and planes can be perpendicular or parallel to lines. Circles and spheres have the same defi ning property, except spheres are in three dimensions. Refl ections and refl ection-symmetry are defi ned the same way in three dimensions as in two except that the refl ecting line is replaced by a refl ecting plane.

Vocabulary

Lesson 9-1

angle formed by a line and a plane

*line perpendicular to a plane

foot of a segmentparallel planesdistance between parallel

planesdistance to a plane from

a point*skew linesdihedral angleedge of a dihedral angleperpendicular planes

Lesson 9-2

surface, solidinterior, exterior of a surfaceface, edge, vertices of a

surface*cylindrical solidbases of a cylindrical solidlateral surfacelateral face of a prismcylindrical surfaceheight, altitude of a solidright solid, oblique solid*cylinder, *prismregular prism, cubelateral edge of a prism

Lesson 9-3

*conic surface, *conic solidbase, apex of a conic solid*pyramid, *conelateral edges, base edges

of a pyramidfaces, lateral faces of a

pyramidright pyramid, oblique

pyramid, regular pyramidaxis of a coneright cone, oblique cone

(continued on next page)

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586 Three-Dimensional Figures

Chapter 9

Architects use views to describe 3-dimensional fi gures in two dimensions. The diffi culty of representing a 3-dimensional object on a 2-dimensional piece of paper is approached by drawing different views of the object and by drawing in perspective. It is also useful to examine the 2-dimensional net that can be folded to create the solid.

The surface area of a solid is the area of its surface. If the solid is a right cylindrical solid, the lateral area is determined by unwrapping the lateral surface to form a rectangle whose length is the distance around the base and whose height is the height of the solid. This gives the formula L.A. = ph.

If the solid is a regular pyramid, the lateral surface can be divided into congruent triangles. The sum of the areas of these triangles leads to the formula L.A. = 1__

2 �p. The lateral area of a cone is determined by comparing it to a regular pyramid with a large number of faces. The formula is the same as for a regular pyramid, but in this case p represents the circumference of the base of the cone.

Vocabulary

Lesson 9-3 (cont.)

lateral surface, lateral edge of a cone

height, altitude of a pyramid or cone

slant height of a pyramidslant height of a cone

Lesson 9-4

vanishing point

Lesson 9-5

view of a 3-dimensional fi gure

isometric drawing

Lesson 9-6

*sphere*radius, center diameter of

a sphere*great circle of a spherehemisphere*small circle of a sphere*plane section2-dimensional cross

sectionconic sections

Lesson 9-7

perpendicular bisector of a segment (in space)

*refl ection image of a point over a plane

refl ecting plane, plane of refl ection

*congruent fi gures (in space)

*refl ection-symmetric space fi gure

symmetry plane

Lesson 9-8

*polyhedron, polyhedrons, polyhedra

faces, edges, vertices of a polyhedron

tetrahedron, hexahedronconvex polyhedronregular polyhedronnetfrustum of a pyramid

Lesson 9-9

surface area, S.A.lateral area, L.A.

Postulates, Theorems, and Properties

Point-Line-Plane Postulate (Expanded)Unique Line Assumption (p. 518)Number Line Assumption (p. 519)Dimension Assumption (p. 519)Unique Plane Assumption (p. 519)Intersection Assumption (p. 519)

Unique Plane Theorem (p. 520)Line-Plane Perpendicular Theorem

(p. 521)

Lateral Area Formula for Right Cylindrical Solids (p. 571)

Surface Area Formula for Cylindrical Solids (p. 572)

Surface Area Formula for Pyramids and Cones (p. 577)

Lateral Area Formula for Regular Pyramids and Right Cones (p. 579)


Chapter Wrap-Up

Chapter

Self-Test9 Take this test as you would take a test in class. You

will need a calculator. Then use the Selected Answers

section in the back of the book to check your work.

Self-Test 587

1. In how many different planes can one circle be? Explain.

2. Draw two perpendicular planes.

3. A certain right triangular prism has exactly one symmetry plane. Draw the prism and its symmetry plane.

4. Draw a top, side and front view of a 5-step open stepladder.

5. Given these three views of the object pictured below, what might it be?

front view side view

top view

6. If a cone has slant height 20 and height 16, what is the diameter of the base?

7. Given the net of the right pyramid below, fi nd the surface area of the pyramid formed when the net is folded up.

12

12

10

10

10

10 10

10 10

10

8. When the net below is folded into a cube, which face is opposite 3?

1 3 4

5

2

6

9. Draw a net for the cylinder below. Give lengths of all segments and any radii on your net.

3"

12



Chapter 9

10. A small circle of a globe is 6" from the center of the globe. If the circle has area 36π square inches, what is the length of the diameter of the globe?

11. The Pyramid of Cestius is one of the best-preserved ancient buildings in Rome. The base of this regular square pyramid is a square with sides 100 feet. The height is 125 feet. Find the lateral area of the pyramid.

12. Can a lateral face of a prism be a triangle? Why or why not?

13. In the right triangular prism pictured below, BC = 5, AB = 4, and AE = 12. Find the surface area of the prism.

B

A

C

D

E

F

5

4

12

14. Using the prism in Question 13, name

a. its two bases.

b. a lateral face.

c. a lateral edge

15. This hollow air conditioning duct is an open-ended cylinder with diameter 28 inches. If its length is 10 feet, what is the area of the material needed to build it?

16. Draw views of the regular hexagonal prism below as seen from the front, side, and top.

17. a. How many planes of symmetry does a regular heptagonal prism have?

b. How many of these planes are perpendicular to its base?

18. Draw an example of one possible cross section of the Liberty Bell below with:

a. a plane parallel to the ground.

b. a plane perpendicular to the ground.

In 19 and 20, redraw the fi gure below

19. in one-point perspective.

20. in two-point perspective.


Chapter Wrap-Up

Chapter Chapter

Review

Chapter Review 589

9

SKILLS Procedures used to get answers

OBJECTIVE A Draw common 3-dimensional

fi gures. (Lessons 9-1, 9-2, 9-3, 9-6)

In 1−7, draw each fi gure.

1. two parallel planes intersected by a line at a 30º angle

2. a line that is perpendicular to a plane

3. a right prism with a nonconvex base

4. an oblique cone

5. two perpendicular planes

6. a sphere with a plane section through its center

7. a cube and a right pyramid whose base is one of the sides of the cube

OBJECTIVE B Give views of a fi gure from

the top, sides, or bottom. (Lesson 9-5)

8. Give each view of the regular square pyramid below.

a. top b. front c. right

9. Give each view of the oblique cylinder below.


front

10. Give each view of the house below.


OBJECTIVE C Calculate surface areas and

lengths in prisms, cylinders, pyramids, and

cones. (Lessons 9-2, 9-3, 9-9, 9-10)

11. The fi gure below shows a cube with sides of length 3.

a. Calculate HB.

b. Calculate HD.

c. Calculate the lateral area of the cube.

BA

G

EF

H

DC

SKILLS

PROPERTIES

USES

REPRESENTATIONS



12. In a regular pentagonal pyramid, the length of each side of the base is 1.5 cm, and the slant height is 3.5 cm.

a. Find the lateral area of the pyramid.

b. Find the area of a lateral face.

In 13 and 14, refer to the regular hexagonal

prism below.

13. If the height of the prism is 4, and the length of the side of a base is 1, fi nd the lateral area.

14. If the height of the prism is 5, and the lateral area is 32, fi nd the surface area of the prism.

15. A right cone has a slant height of 30 cm, and its lateral area is 400 cm2. Find the radius of its base.

16. Find the surface area of the oblique square prism below.

3 3

4

OBJECTIVE D Make and analyze

perspective drawings. (Lesson 9-4)

In 17 and 18, draw the fi gure in perspective.

17.

18.

In 19 and 20, an image is given in perspective. Trace

its outline, and show a vanishing point.

19.

20.

Chapter 9


Chapter Wrap-Up

Chapter Review 591

PROPERTIES Principles behind the mathematics

OBJECTIVE E Make conclusions based on

the Point-Line-Plane Postulate. (Lesson 9-1)

21. Three points lie on exactly one plane. What can you say about those points?

22. Line � lies in plane X.

a. How many different planes are there through �?

b. How many of these are perpendicular to X?

23. The intersection of � �� PQ and plane A contains at least two points. What can you say about the relationship between the points and the plane?

24. Can two planes intersect in exactly two points? Why or why not?

OBJECTIVE F Identify parts of common

3-dimensional fi gures. (Lessons 9-2, 9-3)

25. Identify the requested part of the fi gure below.

a. the altitude

b. the base

c. a lateral face

B

C

D

E

F

A

OI

26. In a cube, how many different pairs of faces could be bases of the cube?

27. How many different pairs of faces could be bases for a regular hexagonal prism?

28. The base of a pyramid is an n-gon. How many faces does the pyramid have?

29. Draw a truncated cone.

OBJECTIVE G Distinguish 3-dimensional

fi gures by their defi ning properties. (Lessons

9-2, 9-3, 9-6)

30. True or False No pyramid can have a pair of parallel faces.

31. True or False Three small circles on the same sphere can have the same radius.

32. In one polyhedron, all of the sides are parallelograms. In the other, all of them are triangles. If one is a pyramid and the other is a prism, which one is which?

33. True or False Any polyhedron whose faces are all rectangles must be a box.

34. A 3-dimensional fi gure you saw in this chapter is defi ned as the set of all points in space equidistant from a given point. What is this fi gure?

OBJECTIVE H Determine symmetry planes

in 3-dimensional fi gures. (Lesson 9-7)

35. At least how many planes of symmetry does a box have?

36. When does a pyramid have a plane of symmetry?

37. How many planes of symmetry does a regular prism with a 12-gon base have?



Chapter 9

In 38 and 39, tell how many symmetry planes the

fi gure has.

38.

triangular prism placed on a cube

39.

a right cone placed on a right cylinder

USES Applications of mathematics in real-world situations.

OBJECTIVE I Draw plane sections of real

life 3-dimensional objects. (Lesson 9-6)

In 40 and 41, refer to the picture at

the right of a tuning fork. Describe

where to place a plane so that its

intersection with the tuning fork will

be the given shape.

40. two disjoint rectangles

41. a circle

In 42 and 43, suppose someone is standing on level

ground wearing the baseball cap shown below. Draw

an example of one possible section of the hat with

the given plane.

42. a plane perpendicular to the ground

43. a plane parallel to the ground

44. The picture below shows some sections of oranges. For one of these oranges, draw what a section would have looked like if the orange were cut in a plane that is perpendicular to the shown section.

OBJECTIVE J Apply formulas for lateral

and surface area to real situations.

(Lessons 9-9, 9-10)

45. A can of vegetables is a right cylinder. The radius of its base is 5 cm and its surface area is 160π cm2. How tall is the can?

46. A paper holder for an ice cream cone is in the shape of a right cone. If its slant height is 8 cm and its radius is 2.5 cm, what is its lateral area?

47. The sides of the base of an ancient square pyramid are 80 cubits (an ancient unit). The pyramid is 40 cubits high. Find its lateral area.


Chapter Wrap-Up

Chapter Review 593

48. Shayna and Regina each have a box to decorate. The dimensions of Shayna’s box are twice the dimensions of Regina’s. Shayna claims that the surface area of her box is twice the surface area of Regina’s box. Regina disagrees. Who is right? Explain your answer.

REPRESENTATIONS Pictures, graphs, or objects that illustrate concepts

OBJECTIVE K From a net, make a surface,

and vice versa. (Lesson 9-8)

49. Tell whether each fi gure below could be a net for a cube.

a.

c.

b.

d.

50. Draw a net for a hexagonal right pyramid.

51. Draw a net for a regular octagonal prism with height 4 and base with edge 5.

52. Name the surface that could be made from the net below.

53. Draw a net for a cylinder with height 7 and base with radius 4.

OBJECTIVE L From 2-dimensional views

of a fi gure, determine the 3-dimensional

fi gure. (Lesson 9-5)

In 54 and 55, use the given views of the buildings.

a. How many stories tall is the building?

b. How many sections long is the building from front

to back?

c. Where is the tallest part of the building located?

54.

front view right view top view

55.

front view left view top view

56. Kendall saw the front and right side views of a cake, shown below. Draw two different possible top views of this cake.

front view right view

57. a. What common object is pictured below?

b. Draw the side view for this object.

front view top view


chapter wrap-up chapter summary and 9 vocabulary

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