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12 Surface Area and Volume 12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids and Cones 12.6 Surface Areas and Volumes of Spheres 12.7 Spherical Geometry EarthEarth (p.(p 692) TeTennisnnis BallsBalls (p(p.. 68685)5) KhKhafre'sf 'P Pyramid id ((p. 674)674) SEE the Big Idea GrGreateat BlBlueue HHoleole (p(p.. 66669)9) TTrTrafficafffic CCoConene ((p(p.. 65656)6) Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Finding the Area of a Circle (7.9.B) Example 1 Find the area of the circle. A = πr2 Formula for area of a circle 8 in. = π ⋅ 82 Substitute 8 for r. = 64π Simplify. ≈ 201.06 Use a calculator. The area is about 201.06 square inches. Find the area of the circle. 1. 2. 3. 20 cm 6 m 9 ft Finding the Area of a Composite Figure (7.9.C) Example 2 Find the area of the composite figure. 17 ft The composite figure is made up of a rectangle, a triangle, and a semicircle. 16 ft Find the area of each figure. 32 ft 30 ft Area of rectangle Area of triangle Area of semicircle 1 πr2 A =ℓw A = — bh A = — 2 2 1 π(17)2 = 32(16) = — (30)(16) = — 2 2 = 512 = 240 ≈ 453.96 So, the area is about 512 + 240 + 453.96 = 1205.96 square feet. Find the area of the composite figure. 4. 7 m 5. 6 in. 6. 6 cm 7 m 3 in. 6 cm 10 in. 6 cm 7 m 5 in. 3 3 cm 20 m 9 cm 7. ABSTRACT REASONING A circle has a radius of x inches. Write a formula for the area of the circle when the radius is multiplied by a real number a. 637 MMathematicalathematical Mathematically profi cient students create and use representations to TThinkinghinking organize, record, and communicate mathematical ideas. (G.1.E) Creating a Coherent Representation CCoreore CConceptoncept Nets for Three-Dimensional Figures A net for a three-dimensional fi gure is a two-dimensional pattern that can be folded to form the three-dimensional fi gure. base w w w h lateral lateral lateral lateral h face face face face w base Drawing a Net for a Pyramid Draw a net of the pyramid. 20 in. 19 in. 19 in. SOLUTION The pyramid has a square base. Its four lateral faces are congruent isosceles triangles. 19 in. 20 in. 19 in. MMonitoringonitoring PProgressrogress Draw a net of the three-dimensional fi gure. Label the dimensions. 1. 2. 3. 5 m 4 ft 15 in. 12 m 4 ft 8 m 10 in. 2 ft 10 in. 638 Chapter 12 Surface Area and Volume 12.1 Three-Dimensional Figures EEssentialssential QQuestionuestion What is the relationship between the numbers TEXAS ESSENTIAL of vertices V, edges E, and faces F of a polyhedron? KNOWLEDGE AND SKILLS G.10.A A polyhedron is a solid that is bounded edge by polygons, called faces. • Each vertex is a point. face • Each edge is a segment of a line. • Each face is a portion of a plane. vertex Analyzing a Property of Polyhedra Work with a partner. The fi ve Platonic solids are shown below. Each of these solids has congruent regular polygons as faces. Complete the table by listing the numbers of vertices, edges, and faces of each Platonic solid. tetrahedron cube octahedron dodecahedron icosahedron Solid Vertices, V Edges, E Faces, F tetrahedron cube octahedron MAKING dodecahedron MATHEMATICAL ARGUMENTS icosahedron To be profi cient in math, you need to reason inductively about data. CCommunicateommunicate YourYour AnswerAnswer 2. What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? (Note: Swiss mathematician Leonhard Euler (1707–1783) discovered a formula that relates these quantities.) 3. Draw three polyhedra that are different from the Platonic solids given in Exploration 1. Count the numbers of vertices, edges, and faces of each polyhedron. Then verify that the relationship you found in Question 2 is valid for each polyhedron. Section 12.1 Three-Dimensional Figures 639 12.1 Lesson WWhathat YYouou WWillill LLearnearn Classify solids. Describe cross sections. Core VocabularyVocabulary Sketch and describe solids of revolution. polyhedron, p. 640 face, p. 640 Classifying Solids edge, p. 640 A three-dimensional fi gure, or solid, is bounded by vertex, p. 640 face fl at or curved surfaces that enclose a single region cross section, p. 641 of space. A polyhedron is a solid that is bounded by solid of revolution, p. 642 polygons, called faces. An edge of a polyhedron is a axis of revolution, p. 642 line segment formed by the intersection of two faces. Previous A vertex of a polyhedron is a point where three or more vertex solid edges meet. The plural of polyhedron is polyhedra edge prism or polyhedrons. pyramid cylinder cone CCoreore CConceptoncept sphere Types of Solids base Polyhedra Not Polyhedra prism cylinder cone pyramid sphere Pentagonal prism To name a prism or a pyramid, use the shape of the base. The two bases of a prism are congruent polygons in parallel planes. For example, the bases of a pentagonal Bases are prism are pentagons. The base of a pyramid is a polygon. For example, the base of a pentagons. triangular pyramid is a triangle. Classifying Solids Triangular pyramid Tell whether each solid is a polyhedron. If it is, name the polyhedron. a. b. c. Base is a triangle. SOLUTION a. The solid is formed by polygons, so it is a polyhedron. The two bases are congruent rectangles, so it is a rectangular prism. b. The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. c. The cone has a curved surface, so it is not a polyhedron. 640 Chapter 12 Surface Area and Volume MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Tell whether the solid is a polyhedron. If it is, name the polyhedron. 1. 2. 3. Describing Cross Sections Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section. For example, three different cross sections of a cube are shown below. square rectangle triangle Describing Cross Sections Describe the shape formed by the intersection of the plane and the solid. a. b. c. d. e. f. SOLUTION a. The cross section is a hexagon. b. The cross section is a triangle. c. The cross section is a rectangle. d. The cross section is a circle. e. The cross section is a circle. f. The cross section is a trapezoid. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Describe the shape formed by the intersection of the plane and the solid. 4. 5. 6. Section 12.1 Three-Dimensional Figures 641 Sketching and Describing Solids of Revolution A solid of revolution is a three-dimensional fi gure that is formed by rotating a two-dimensional shape around an axis. The line around which the shape is rotated is called the axis of revolution. For example, when you rotate a rectangle around a line that contains one of its sides, the solid of revolution that is produced is a cylinder. Sketching and Describing Solids of Revolution Sketch the solid produced by rotating the fi gure around the given axis. Then identify and describe the solid. a. 9 b. 4 4 9 5 2 SOLUTION a. 9 b. 4 5 2 The solid is a cylinder with a height of 9 and a base The solid is a cone with radius of 4. a height of 5 and a base radius of 2. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Sketch the solid produced by rotating the fi gure around the given axis. Then identify and describe the solid. 7. 8. 6 9. 7 3 7 8 8 4 6 642 Chapter 12 Surface Area and Volume 12.1 Exercises Tutorial Help in English and Spanish at BigIdeasMath.com VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. VOCABULARY A(n) __________ is a solid that is bounded by polygons. 2. WHICH ONE DOESN’T BELONG? Which solid does not belong with the other three? Explain your reasoning. MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics In Exercises 3–6, match the polyhedron with its name. In Exercises 11−14, describe the cross section formed by the intersection of the plane and the solid. 3. 4. (See Example 2.) 11. 12. 5. 6. 13. 14. A. triangular prism B. rectangular pyramid C. hexagonal pyramid D. pentagonal prism In Exercises 7–10, tell whether the solid is a polyhedron. If it is, name the polyhedron. (See Example 1.) In Exercises 15–18, sketch the solid produced by 7. 8. rotating the fi gure around the given axis. Then identify and describe the solid. (See Example 3.) 15. 8 16. 8 8 6 9. 10. 8 6 17. 18. 5 3 2 2 3 5 Section 12.1 Three-Dimensional Figures 643 19. ERROR ANALYSIS Describe and correct the error in 28. ATTENDING TO PRECISION The fi gure shows a plane identifying the solid. intersecting a cube through four of its vertices. The ✗ edge length of the cube is 6 inches. The solid is a rectangular pyramid. a. Describe the shape formed by the cross section. 20. HOW DO YOU SEE IT? Is the swimming pool shown b.
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