11-7 lesson three-dimensional figures notes -...

MAIN IDEABuild three-dimensional figures given the top, side, and front views.

New Vocabularythree-dimensional

figurefaceedgelateral facevertex (vertices)prismbasepyramidconecylinderspherecenter

Math Onlineglencoe.com• Extra Examples• Personal Tutor• Self-Check Quiz

Key ConceptPrisms and PyramidsFigure Properties

Prism

• Has at least three lateral faces that are parallelograms.

• The top and bottom faces, called the bases, are congruent parallel polygons.

• The shape of the base tells the name of the prism.

Rectangularprism

Square prism or cube

Pyramid

• Has at least three lateral faces that are triangles.

• Has only one base, which is a polygon.

• The shape of the base tells the name of the pyramid.Triangular pyramid

Squarepyramid

11-7

Study the shape of each common object below. Then compare and contrast the properties of each object.

Many common shapes are three-dimensional figures. That is, they have length, width, and depth (or height). Some terms associated with three-dimensional figures are shown below.

A face is a flat surface.

The sides are called lateral faces.

The edges are the segments formed by intersecting faces.

The edges intersect at the vertices.

Two types of three-dimensional figures are prisms and pyramids.

See Ch. 11 Answer Appendix.See Ch. 11 Answer Appendix.

Three-Dimensional Figures

Lesson 11-7 Three-Dimensional Figures 603

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11-7 LessonNotes

1 Focus

Vertical Alignment

Before Lesson 11-7Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids

Lesson 11-7Classify three-dimensional figures

After Lesson 11-7Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres

2 Teach

Scaffolding QuestionsAs you ask the following questions, point to a chalkboard eraser, cardboard box, or other rectangular prism.

Ask:• What shape are the top and the

bottom? rectangle

• Are the top and bottom parallel? yes

• How many sides does it have? 4

• Are the sides flat? yes

• How many vertices does it have? 8

Prisms

Make sure students realize that the bases of a prism can be positioned toward the sides. Also make sure they realize that any pair of opposite faces of a rectangular or square prism can be the bases.

Differentiated InstructionVisual/Spatial Learners While presenting the Key Concept box, have students create nets for a rectangular prism, triangular prism, square prism, triangular pyramid, and square pyramid to make visual connections to the properties for each figure. For example, by making a net for a square pyramid, students can visualize the three triangular surfaces and the one square surface of the pyramid.


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Key ConceptCones, Cylinders, and Spheres

Figure Properties

Cone

• Has only one base.

• The base is a circle.

• Has one vertex.

Cylinder

• Has only two bases.

• The bases are congruent circles.

• Has no vertices and no edges.

Sphere

• All of the points on a sphere are the same distance from the center.

• No faces, bases, edges, or vertices.

center

Some three-dimensional figures have curved surfaces.

Classify Three-Dimensional Figures

For each figure, identify the shape of the base(s). Then classify the figure.

1 The figure has one circular base, no edge, and one vertex.

2 The base and all other faces are squares.

The figure is a cone. The figure is a square prism or cube.

a. b.

3 CAMERAS Classify the shape of the body of the digital camera, not including the lens, as a three-dimensional figure.

The body of the camera is a rectangular prism.

c. Classify the shape of the zoom lens as a three-dimensional figure.

triangle; pyramidtriangle; pyramid circle; cylindercircle; cylinder

cylindercylinder

Prisms and Pyramids Prisms and Pyramids Prisms and pyramids are examples of polyhedra, or solids with flat surfaces that are polygonal regions. Cones, cylinders, and spheres are not examples of polyhedra.

Three-Dimensional Figures Three-Dimensional Figures In three-dimensional figures, dashed lines are used to indicate edges that are hidden from view.

604 Chapter 11 Two- and Three-Dimensional Figures

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Prisms and pyramids are classified by the shape and number of their bases.

Cones and cylinders have curved surfaces that do not intersect (with the base or bases) to form segments, or edges.

Many everyday objects approximate the shapes of three-dimensional geometric figures.

Focus on Mathematical Content

Formative Assessment

Use the Check Your Progress exercises after each Example to determine students’ understanding of concepts.

ADDITIONAL EXAMPLES

Additional Examples are also in:

• Noteables™ Interactive Study Notebook with Foldables™

• Interactive Classroom PowerPoint® Presentations

Classify each figure.

rectangular pyramid

rectangular prism

HOUSES Classify the shape of the house’s roof as a three-dimensional figure.

triangular prism

604 Chapter 11 Measurement: Two- and Three-Dimensional Figures

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ForExercises

5–89–10

SeeExamples

1–23

HOMEWORK HELPHELP

See pages 699, 714.EXTRA PRACTICEPRACTICE

Examples 1, 2(p. 604)


1. 2. 3.

Example 3(p. 604)

4. SPORTS An official major league baseball has 108 stitches. Classify the shape of a baseball as a three-dimensional figure. sphere


5. 6. 7. 8.

9. FOOD What three-dimensional figure describes the item at the right? cone

10. SCHOOL SUPPLIES Classify the shape of your math textbook as a three-dimensional figure. rectangular prism


11. 12. 13.

14. SCHOOL SUPPLIES The model of the pencil shown is made of two geometric figures. Classify these figures.

15. HOUSES The model of the house shown is made of two geometric figures. Classify these figures. triangular prism and rectangular prism

square; square pyramidsquare; square pyramid triangle; triangular prism triangle; triangular prism

circle; cylindercircle; cylinder

Exercise LevelsA: 5–10B: 11–15C: 16–19

Exercise LevelsA: 5–10B: 11–15C: 16–19

triangle; triangular pyramidtriangle; triangular pyramid

6. triangle; triangular prism6. triangle; triangular prism

7. rectangle; rectangular pyramid

7. rectangle; rectangular pyramid

8. square; square prism or cube8. square; square prism or cube

11–13. See margin.11–13. See margin.

cylinder and conecylinder and cone


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Prisms and Pyramids

Three-Dimensional Figures

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3 Practice

Formative Assessment

Use Exercises 1–4 to check for understanding.

Then use the chart at the bottom of this page to customize your assignments for students.

Intervention You may wish to use the Study Guide and Intervention Master on page 47 of the Chapter 11 Resource Masters for additional reinforcement.

Odd/Even AssignmentsExercises 5–10 are structured so that students practice the same concepts whether they are assigned odd or even problems.

Additional Answers 11. trapezoid; trapezoidal prism

12. pentagon; pentagonal pyramid

13. octagon; octagonal prism

HOMEWORK OPTIONS

Level Assignment Two-Day Option

Basic 5–10, 16, 18–29 5–9 odd, 20, 21 6–10 even, 16, 18, 19, 22–29

Core 5–13 odd, 14–16, 18–29 5–10, 20, 21 11–16, 18, 19, 22–29

Advanced/Pre-AP 11–26 (optional: 27–29)

DIFFERENTIATED


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20. Which statement is true about all triangular prisms? C

A All of the edges are congruent line segments.

B There are exactly 6 faces.

C The bases are congruent triangles.

D All of the faces are triangles.

21. Which figure is shown? F

F triangular pyramid

G square pyramid

H rectangular pyramid

J triangular prism

22. MEASUREMENT Find the area of the figure shown at the right if each triangle has a height of 3.5 inches and the square has side lengths of 4 inches. (Lesson 11-6) 44 in 2

23. MEASUREMENT Find the area of a circle with a radiusof 5.7 meters. Round to the nearest tenth. (Lesson 11-4) 102.1 m 2

ALGEBRA Find the missing angle measure in each quadrilateral. (Lesson 10-6)

24. x°

123°70°

25.

x°

92°

87°

128°

26.

x°100°

94°68°

PREREQUISITE SKILL Describe the shape seen when each object is viewed from the top.

27. number cube square 28. cereal box rectangle 29. soup can circle

77°77° 53°53° 98°98°

H.O.T. Problems 16. REASONING Two sets of figures were sorted according to a certain rule. The figures in Set A follow the rule and the figures in Set B do not follow the rule. Describe the rule.

Set A Prism Pyramid Cube

Set B Cylinder Cone Sphere

17. CHALLENGE What figure is formed if only the height of a cube is increased? Draw a figure to justify your answer.

18. OPEN ENDED Select one three-dimensional figure in which you could use the term congruent to describe the bases of the figure. Then write a sentence using congruent to describe the figure. See margin.

19. MATHWRITING IN Apply what you know about the properties of geometric figures to compare and contrast cones and pyramids. See margin.

16. Set A is a list of figures that are made up of flat surfaces only and set B is a list of figures that have curved surfaces.

16. Set A is a list of figures that are made up of flat surfaces only and set B is a list of figures that have curved surfaces.

rectangular prism; See students’ work for justification. rectangular prism; See students’ work for justification.


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4 Assess

Ticket Out the Door Have students draw an everyday object that resembles a three-dimensional geometric figure. Have them classify the object according to the figure it resembles, pointing out its properties.

Foldables™Follow-Up

Remind students to take notes about the classification of three-dimensional figures in their Foldables. Encourage them to give examples of each figure, noting its properties.

Additional Answers 18. Sample answer: The measures of

the areas of the bases of a rectangular prism are equal.

19. Sample answer: A cone has only one base that is a circle. A pyramid also has only one base but its base is a polygon. They both have only one vertex. A cone does not have any lateral faces and a pyramid has at least three lateral faces.

Pre-AP Activity Use after Exercise 18

Have students write a formula for the total area of the faces of a cube with sides of length x. A = 6x2


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11-7 lesson three-dimensional figures notes -...

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