MAIN IDEA Build three-dimensional figures given the top, side, and front views. New Vocabulary three-dimensional figure face edge lateral face vertex (vertices) prism base pyramid cone cylinder sphere center Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz Key Concept Prisms and Pyramids Figure 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. Rectangular prism 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 Square pyramid 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 11-7 Lesson Notes 1 Focus Vertical Alignment Before Lesson 11-7 Visualize and draw two- dimensional views of three- dimensional objects made from rectangular solids Lesson 11-7 Classify three-dimensional figures After Lesson 11-7 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres 2 Teach Scaffolding Questions As 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 Instruction Visual/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. Lesson 11-7 Three-Dimensional Figures 603 603_0603_0606_CH11_874048 603 603_0603_0606_CH11_874048 603 11/28/07 5:52:22 PM 11/28/07 5:52:22 PM
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MAIN IDEABuild three-dimensional figures given the top, side, and front views.
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.
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.
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.
605_0603_0606_CH11_874048 605605_0603_0606_CH11_874048 605 11/19/07 11:48:25 AM11/19/07 11:48:25 AM
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.
606 Chapter 11 Measurement: Two- and Three-Dimensional Figures
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
606 Chapter 11 Measurement: Two- and Three-Dimensional Figures
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