Volumes Of Pyramids 11-5 And Cones
Volumes of Pyramidsand Cones11-511-51. PlanGO for HelpWhat You’ll LearnCheck Skills You’ll Need To find the volume of aUse the Pythagorean Theorem to find the value of the variable.pyramid To find the volume of acone1.h. . . And WhyTo find the volume of astructure in the shape of apyramid, as in Example 12.15 cmᐉ2m9 cm12 in.11232.5 m41To find the volume ofa pyramidTo find the volume of a coneExamples1.5 m8 in.Objectives23.10 in.h12 cmLesson 8-1Real-World ConnectionFinding Volume of a PyramidFinding Volume of an ObliqueConeReal-World ConnectionMath BackgroundFinding Volume of a PyramidA hands-on activity will relate thevolume of a prism to the volumeof three pyramids with the samebases and heights. Students willgeneralize this relationship andrelate the volume of a cylinder tothe volume of three cones withthe same bases and heights.Euclid included the proof of thelatter relationship in Book XII ofThe Elements. Later, Archimedesattributed the proof to Eudoxusand his “method of exhaustion.”This method is the forerunner ofthe modern method of limits.Hands-On Activity: Finding VolumeYou know how to find the volume of a prism. Use the following toexplore finding the volume of a pyramid. Draw the netsshown at the righton cardboard. Cut out the netsand tape themtogether to make acube and a regularsquare pyramid.Each model willhave one open face.5 cm5.6 cm5 cmMore Math Background: p. 596DThe areas are .1. How do the areas of the bases of the cube and the pyramid compare?2. How do the heights of the cube and pyramid compare? The heightsare about .3. Fill the pyramid with rice or other material. Then pour the rice fromthe pyramid into the cube. How many pyramids full of rice does thecube hold? about 3 pyramids4. The volume of the pyramid is what fractional part of the volume ofthe cube? 13Lesson Planning andResourcesSee p. 596E for a list of theresources that support this lesson.PowerPointBell Ringer PracticeThe volume of a pyramid is a particular fraction of the volume of a prism that hasthe same base and height as the pyramid. The fraction is shown and this fact isstated as Theorem 11-8 at the top of the next page.Lesson 11-5 Volumes of Pyramids and ConesSpecial NeedsL1Have students repeat “the area of a triangle is 21 thearea of a rectangle in two-dimensions, and thevolume of a pyramid is 12 the volume of a prism inthree-dimensions.”learning style: verbalBelow Level631Check Skills You’ll NeedFor intervention, direct students to:The Pythagorean TheoremLesson 8-1: Example 1Extra Skills, Word Problems, ProofPractice, Ch. 8L2Write Theorems 11-8 and 11-9 on the board.Have students explain how they are alike and howthey are different.learning style: visual631
2. TeachKey ConceptsTheorem 11-8Volume of a PyramidThe volume of a pyramid is one third the product ofthe area of the base and the height of the pyramid.Guided Instructionh1V 3 BhBHands-on ActivityAfter students tape their nets,have them estimate the ratioof the pyramid’s volume to thecube’s volume. They may besurprised to find the ratio is 13.2EXAMPLEBecause of Cavalieri’s Principle, the volume formulais true for all pyramids, including oblique pyramids.The height h of an oblique pyramid is the length ofthe perpendicular segment from the vertex to theplane of the base.GO for HelpYou can findCavalieri’s Principleon page 625.Teaching TiphOblique PyramidHave students explain why oneleg of the right triangle is 20 ft.1EXAMPLEReal-WorldConnectionArchitecture The Pyramid is an arena inMemphis, Tennessee. The area of the baseof The Pyramid is about 300,000 ft 2.Its height is 321 ft. What is the volumeof The Pyramid?Use the formula forV 13Bhvolume of a pyramid.PowerPointAdditional Examples1 Find the volume of a squarepyramid with base edges 15 cmand height 22 cm. 1650 cm3 13(300,000)(321) Substitute.2 Find the volume of the square 32,100,000pyramid with base edges 16 mand slant height 17 m.Simplify.The volume is about 32,100,000 ft3.Quick Check17 m1 Find the volume of a square pyramid with base edges 12 in. and height 8 in.384 in.3hTo find the volume of a pyramid you may first need to find its height.16 m216 m81280 MPLEFinding Volume of a PyramidGridded Response Find the volume in cubic feet of a square pyramid withbase edges 40 ft and slant height 25 ft.Step 1: Find the height of the pyramid. h2 625 h2 40025220 2h 2 225h 15Use the Pythagorean Theorem.Simplify.25 fth40 ftSolve for h2.Take square roots.hStep 2: Find the volume of the pyramid.V 13 Bh 13(40 ? 40)15 800025 ft20 ftUse the formula for volume of a pyramid.Substitute.Simplify.The volume of the pyramid is 8000 ft3.Quick Check6322 Find the volume of a square pyramid with base edges 24 m and slant height 13 m.960 m3Chapter 11 Surface Area and VolumeAdvanced LearnersEnglish Language Learners ELLL4Ask students to write a formula for the volume of asquare pyramid whose base edge measures s and slant2height measures h.V 5 s "4h2 2 s26632learning style: verbalWatch for students who confuse slant height withheight in the formula for the volume of a pyramid.Reinforce the relationship between the volume ofa pyramid and the volume of a prism, which alsouses height.learning style: verbal
Guided Instruction21Finding Volume of a Cone3You have seen that the volume of a pyramid is one third the volume of a prismwith the same base and height. Similarly, the volume of a cone is one third thevolume of a cylinder with the same base and height.The altitude of a right coneintersects the center of the base.Remind students that althoughthe base of an oblique cone isstill circular, the altitude of anoblique cone does not intersectthe center of the base, and maynot even intersect the base.4For: Pyramid, Cone ActivityUse: Interactive Textbook, 11-5Theorem 11-9EXAMPLEDiversityPoint out that not all NativeAmericans lived in teepees.Encourage students to investigateother dwellings used by differenttribes.The cones and the cylinder have the same base and height.It takes three cones full of rice to fill the cylinder.Key ConceptsEXAMPLEVolume of a ConePowerPointThe volume of a cone is one third the product ofthe area of the base and the height of the cone.hV 13 Bh, or V 13 pr 2hAdditional ExamplesB3 Find the volume of the cone interms of p.3A145AABBEDEDCBEDCBA3CBA2CCFinding Volume of an Oblique ConeEXAMPLEEDCB11 in.This volume formula applies to all cones, including oblique cones.DDEETest-Taking TipThe formula V 5 13 Bhcan be used to findthe volume of apyramid or a cone.The only difference ishow you calculate B,the area of the base.Find the volume of an oblique cone with diameter 30 ftand height 25 ft. Give your answer in terms of p and alsorounded to the nearest cubic foot.V 13 pr 2h 13 p(15)2 2533π in.3Use the formula for volume of a cone.4 An ice cream cone is 7 cmtall and 4 cm in diameter. Abouthow much ice cream can fitentirely inside the cone? Findthe volume to the nearest wholenumber. 29 cm330 ftSubstitute 15 for r and 25 for h. 1875p 6 in.25 ftSimplify.5890.4862Use a calculator.The volume of the cone is 1875p ft3, or about 5890 ft3.Quick Check3 Find the volume of each cone in terms of p and also rounded as indicated.a. to the nearest cubic meterb. to the nearest cubic millimeter144π m3; 452 m312 m6mResources Daily Notetaking Guide 11-5L3342 mm 6174π mm 3;19,396 mm Daily Notetaking Guide 11-5—L1Adapted Instruction21 mmClosureA cone-shaped structure can be particularly strong, as downward forces at thevertex are distributed to all points in its circular base.Lesson 11-5 Volumes of Pyramids and Cones633Explain how to find the volume ofa cone if you know its height andthe circumference of its base.Use C 2πr to find r, B πr2to find B, and V 31Bh to find V,or use C 2πr to find r and thenV 13πr2h to find V.633
3. Practice4Real-WorldConnectionDwelling The covering on a teepee rests on poles that come together likeconcurrent lines. The resulting structure approximates a cone. If the teepeepictured is 12 ft. high with a base diameter 14 ft, estimate its volume.Assignment Guide1 A B 1-10, 16-18, 20, 21, 24V 13 pr 2hV 13 p(7)2(12)2 A B11-15, 19, 22, 23,25-30C Challenge31-36Test PrepMixed ReviewEXAMPLE Use the formula for the volume of a cone.Substitute; r 12 (14) 7.615.75216Use a calculator.The volume of the teepee is approximately 616 ft3.37-4243-48Quick Check4 A small child’s teepee is 6 ft tall and 7 ft in diameter. Find the volume of the teepeeto the nearest cubic foot. 77 ft 3Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 6, 14, 19, 20, 24.Exercise 1 Point out that it isunnecessary to know the shapeof the pyramid’s base becausethe area of the base is given.EXERCISESAPractice by ExampleExample 1GO forHelpError Prevention!Exercises 8–10 Students mayFor more exercises, see Extra Skill, Word Problem, and Proof Practice.Practice and Problem Solving(page 632)1. The entrance to the Louvre Museum in Paris, France is a pyramid. The heightof the pyramid is about 70 ft and the area of its base is about 10,000 ft2. What isthe volume of the pyramid? about 233,333 ft3Find the volume of a square pyramid with the following dimensions.2. base edges 10 cm,height 6 cm 200 cm3substitute the slant height forh in the volume formula. Remindthem to use the PythagoreanTheorem to find h.3. base edges 18 in.,height 12 in. 1296 in.3Find the volume of each square pyramid.about 443.7 cm35.6.11 cm9 in.4. base edges 5 m,height 6 m 50 m37.24 m11 cm10 in.11 cmExample 2GPS Guided Problem SolvingL3L2ReteachingL1Adapted PracticePracticeNameClassL3DatePractice 11-5Example 3Circles in the Coordinate PlaneFind the center and radius of each circle.3.(page 633)2. (x - 2)2 (y - 7)2 49(x 1)2 (y 6)2 169.24 mm16 m16 m2048 m310.15 ft12 mL4Enrichment1. x2 y2 368.(page 632)300 in.34.(x 3)2 (y - 11)2 1223 mm11 ft10 mabout 3714.5 mm3about562.9 ft3about 363.6 m3Find the volume of each cone in terms of π and also rounded as indicated.11. nearest cubic foot12. nearest cubic inch13. nearest cubic inchWrite the standard equation of each circle.5. center (0, 0); r 76. center (4, 3); r 87. center (5, 3); r 29. center (-2, -5); r "28. center (-5, 4); r 1210. center (-1, 6); r "5Write an equation for each circle.11.y612.6y13.y22ⴚ414.2ⴚ24x8xyⴚ4 ⴚ2ⴚ215.24610ⴚ104 ft44424xⴚ4 ⴚ2ⴚ2y16.10 20 30 40x4ⴚ82 44 fty4ⴚ48x Pearson Education, Inc. All rights reserved.ⴚ6Graph each circle. Label its center, and state its radius.17. x2 y2 2519.(x 2)2 (y 4)2 1618. (x - 3)2 (y - 5)2 920.(x 1)2 (y - 1)2 36Write an equation for each circle with the given center that passes throughthe given point.21. center (0, 0); point (3, 4)22. center (5, 9); point (2, 9)23. center (-4, -3); point (2, 2)24. center (7, -2); point (-1, -6)y6Write an equation that describes the position and range of each circle.425. 䉺B26. 䉺F2ⴚ6 ⴚ4 ⴚ2BF2ⴚ2ⴚ4ⴚ66344 in.ⴚ8ⴚ49 in.15 2 in.x468x163π634ft 3; 17 ft3Chapter 11 Surface Area and Volume223 πin.3; 23 in.37 in.36.75π in.3; 115 in.3
Example 4(page 634)14. Chemistry In a chemistry lab you use a filter paper cone to filter a liquid. Thediameter of the cone is 6.5 cm and its height is 6 cm. How much liquid will thecone hold when it is full? about 66.4 cm315. Chemistry This funnel has a filter that was being usedto remove impurities from a solution but becameclogged and stopped draining. The remaining solution isrepresented by the shaded region. How many cubiccentimeters of the solution remain in the funnel?about 4.7 cm3BApply Your SkillsFind the volume to the nearest whole number.312 cm316. 123 in.317.15 cm7.5 in.3 cmExercise 20 Do part b as a classft39 ft15 ft7 in.Square base19. They are equal; bothvolumes are 13πr2h.12 cm24 ft24 ftSquare baseEquilateral basex 2 Algebra Find the value of the variable in each figure. Leave answers in simplest23.3 "246xExercise 2026. cube: 8 units3,cone: 23π units3,pyramid: 83 units3GOnlineHomework HelpVisit: PHSchool.comWeb Code: aue-1105xxVolume 18 37Careershow to construct level floors andvertical walls. Have studentsinvestigate how carpentry toolssuch as plumb bobs and levels useprinciples of geometry to do this.20. Architecture The Transamerica Pyramid in San Francisco (see photo at left)is 853 ft tall with a square base that is 149 ft on each side.a. What is its volume to the nearest thousand cubic feet? 6,312,000 ft3b. How tall would a prism-shaped building with the same square base as thePyramid have to be to have the same volume as the Pyramid?about 284 ftxexercise. Write the formulas forthe volume of a prism and thevolume of a pyramid side by side.Then challenge students to solvethe problem using mental mathand explain their methods.Exercise 24 Builders must know19. Writing The two cylinders pictured at theright are congruent. How does the volume of thelarger cone compare to the total volume of thetwo smaller cones? Explain. See left.radical form. The diagrams are not to scale.21. 622. 3the volumes of two space figures.Ask: What figure is on thebottom? rectangular prismWhat figure is on the top?pyramidConnection to Mental Math2 cm10,36818.Exercise 18 Students must findExercise 27 The answer to part cwill be generalized to all similarcones in Lesson 11-7.Visual LearnersExercise 33 Have students withstrong visual skills explain to therest of the class what the solidlooks like, then cut off the tipof a conical ice cream cone toillustrate it.rVolume 24pVolume 21p24. Hardware Builders use a plumb bob to find a vertical line.2 cmGPS The plumb bob shown combines a regular hexagonal prismwith a pyramid. Find its volume to the nearest cubic centimeter.73 cm325. Multiple Choice A cone has a volume of 600p in.3 and aheight of 50 in. What is the radius of the cone? B3.5 in.6.0 in.10.6 in.36.0 in.6 cm3 cm26. A cone with radius 1 fits snugly inside a square pyramid whichfits snugly inside a cube. What are the volumes of the three figures? See left.27. A cone with radius 3 ft and height 10 ft has a volume of 30p ft3. What is thevolume of the cone formed when the following happens to the original cone?a. The radius is doubled. 120π ft 3 b. The height is doubled. 60π ft 3c. The radius and the height are both doubled. 240π ft 3lesson quiz, PHSchool.com, Web Code: aua-1105Lesson 11-5 Volumes of Pyramids and Cones635635
4. Assess & ReteachPowerPointLesson QuizFind the volume of each figure.When appropriate, leave youranswer in terms of π.1.5 ft29. cone with r 4 andh 3; 16π30. cone with r 3 andh 4; 12π31. cylinder with r 4,h 3, with a cone ofr 4, h 3 removedfrom it; 32π32. cone with r 4, h 513 , with a cone ofr 1, h 113 cut offthe top, and a cylinderof r 1 cut out of itscenter; 24πC6 ftChallenge28. List the volumes of the cone, prism, and pyramid in order from least to greatest.cone: 234.6 in.3; prism: 240 in.3; pyramid: 256 in.38 in.8 in.8 in.10 in.14 in.12 in.4 in.6 in.Visualization The plane region is revolved completelyabout the given line to sweep out a solid of revolution.Describe the solid. Then find its volume in terms of π.29–32. See left.29. the y-axis30. the x-axis31. the line x 4y2x 4 2 O2432. the line y -133. A frustum of a cone is the part that remains whenthe vertex is cut off by a plane parallel to the base.33a. Seehmargin. a. Explain how to use the formula for the volume ofrHa cone to find the volume of a frustum of a cone.b. Containers A 9-in. tall popcorn container is theRfrustum of a cone. Its small radius is 4.5 in. and itslarge radius is 6 in. What is its volume?Frustum of coneabout 784.6 in.334. A disk has radius 10 m. A908 sector is cut away, and10 ma cone is formed. 47.1 ma. What is the circumferenceof the base of the cone?b. What is the area of thebase of the cone? 176.7 m2c. What is the volume of thecone? (Hint: Use the slantheight and the radius of the base to find the height.) 389.6 m36 ft60 ft32.10 mm21 mm1470π mm33. square pyramid with baseedges 24 in. long and slantheight 15 in. 1728 in.34. cone with diameter 3 m andheight 4 m 3π m35.6 ftGraphing Calculator In Exercises 35 and 36, the volume of the solid is 1000 cm3.Use the Activity Lab on page 616 to help you complete each exercise.35. For a square pyramid, find the length of a side of the base for which the lateralarea is as small as possible. about 16.2 cm4 ft36. For a cone, find the radius for which the lateral area is as small as possible.about 8.8 cm5 ft5 ft150 ft3Test PrepAlternative AssessmentHave partners solve the followingexercise and show their work indetail: A cone and a squarepyramid each have height 12 cm.The perimeter of the base of thepyramid and the circumferenceof the base of the cone are each24 cm. Which has the greatervolume?636Multiple Choice37. What is the volume of a 6-ft high square pyramid with base edges 8 ft? AA. 128 ft 3B. 192 ft 3C. 256 ft 3D. 384 ft 338. What is the volume of a cone with diameter 21 m and height 4 m? FF. 147p m3G. 220.5p m3H. 294p m3J. 441p m339. What is the volume of an oblique cone with radius 9 cm and height 12 cm?B. 486p cm3C. 648p cm3D. 972p cm3 AA. 324p cm3636Chapter 11 Surface Area and Volume33. a. The frustum has vol.V 31 πR2H – 13 πr2 h 1223 π(R H – r h). Now ifh1 H – h is thefrustum’s height,V 31 π(R2 (h1 h) – r 2 h) 123 π(R h1 h(R2 – r 2 )).hBy similar , r 5rh1or h R 2 r .h1 1 hR ,Simplifying, V 123 πh1(r rR R2 ).
Test Prep40. What is the volume of the square pyramid at the right? FF. 1568 m3G. 1633 m3H. 2352 m3J. 2450 m341. What is the volume of an oblique square pyramidwith base edges 25 in. and height 24 in.? AA. 5000 in.3B. 7500 in.33C. 10,000 in.D. 15,000 in.3Short Response25 m14 m14 m42. The volume of a cone is 82,418p cm3. Its diameter is 203 cm. What is itsheight? Show all your work, including any formulas that you use.See margin.ResourcesFor additional practice with avariety of test item formats: Standardized Test Prep, p. 657 Test-Taking Strategies, p. 652 Test-Taking Strategies withTransparenciesMixed ReviewGO forHelpLesson 11-443. Sports A cylindrical hockey puck is 1 in. high and 3 in. in diameter. What is itsvolume in cubic inches? Round your answer to the nearest tenth. 7.1 in.344. A triangular prism has height 30 cm. Its base is a right triangle with legs 10 cmand 24 cm. Find the volume of the prism. 3600 cm3Lesson 10-5Lesson 8-245. Find the area of a regular pentagon with a radius 5 in. Give your answer to thenearest tenth of a square inch. 59.4 in.2Find the area of each equilateral triangle to the nearest tenth of a square unit.57.7 in.246. The triangle has 12 cm sides. 62.4 cm2 47. The triangle has 10-in. altitudes.48. Find the area of a 308-608-908 triangle with shorter leg of length 4 cm. 13.9 cm2Geometry at WorkPackage DesignerEach year, more than one trillion dollars in manufactured goods arepackaged in containers. To create each new box, bag, or carton,package designers must balance such factors as safety, environmentalimpact, and attractiveness against cost of production.Consider the three boxesof dishwasher detergent. Allthree boxes have standardvolumes of 108 in.3. The 9 in.6 in.6 in.boxes have different shapes,3 in.however, and different surface4 in.2 in.412 in.6 in.6 in.areas. The box on the left hasthe greatest surface area and therefore costs the most to produce. Despite thehigher cost, the box on the left has become standard. In this case, the leastexpensive package on the right is too difficult for a consumer to pick up and pour.PHSchool.comFor: Information about package designWeb Code: aub-2031Lesson 11-5 Volumes of Pyramids and Cones42. [2] 21 d r;r 21 (203) 101.5;V 123 πr h,so63782,418π 123 π(101.5) h,andh 24 cm.[1] one computationalerror637
632 Chapter 11 Surface Area and Volume Because of Cavalieri’s Principle, the volume formula is true for all pyramids, including oblique pyramids. The height h of an oblique pyramid is the length of the perpendicular segment from the vertex to the plane of the base. Architecture The Pyramid is an arena in Memphis,Tennessee.The area of the base
pyramids under which to bury their pharaohs. This is the so-called Pyramid Age' of Egypt when more than a hundred pyramids were built. In the beginning, the Egyptians constructed step pyramids, then, from the Fourth Dynasty on, they built true pyramids, the biggest of which is the Great Pyramid at Giza. Building a pyramid was the major event during
8.2 Volumes of Cones Work with a partner. You can remember the volume formulas for prisms, cylinders, pyramids, and cones with just two concepts. Volumes of Prisms and Cylinders Volume Area of base Volumes of Pyramids and Cones Volume Volume of
ecological pyramids to size spectra illuminates size-based constraints on the shapes expected for ecological pyramids (as explained below). Conversely, converting from size spectra to ecological pyramids is a powerful method for visualizing the abstract concept of the size spectrum and the underlying parameter combinations (Box 3). Review Trends
Volume and Surface Area of ones and Pyramids Strand: Measurement and Geometry Topic: Solve problems involving volume and surface area of cones and pyramids. Primary SOL: 8.6 The student will a) solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids Related SOL: 8.14, 8.17 Materials
Jan 17, 2017 · Time Table Day/Date Classwork Homework Tuesday 1/3 Introduction Section 1: Mummification Section 1: Mummification Wednesday 1/4 Section 2: Pyramids of Egypt Section 2: Pyramids of Egypt Thursday 1/5 Section 3: Pharaohs of Egypt Section 3: Pharaohs of Egypt Friday 1/6 Rough Drafts Rough Drafts Monday 1/
Power of the Pyramids graph paper Colored pencils Rulers Calculators Key Terms:cohort, demographers, population pyramid population connection 2008 Population Dynamics Student Activity Power of the Pyramids page 1 of 4 9876543210123456789 AGE MALE WORLD FEMALE 75 70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24 15 .
Books Magic Tree House: Mummies in the Morning Magic Tree House Research Guide: Mummies and Pyramids Pyramids: A First Discovery Book I am the Mummy Heb-Nefert Mummies Made in Egypt Secrets of the Mummies Kingfisher Pyramids Into the Mummy’s Tomb Art Build a pyramid
7.Advanced Engineering Mathematics - Chandrika Prasad & Reena Garg 8.Engineering Mathematics - I, Reena Garg . MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY B.Sc. IN NAUTICAL SCIENCE SEMESTER – I BNS 103 NAUTICAL PHYSICS 80 Hrs 1 Heat and Thermodynamics: 15 hrs Heat Transfer Mechanism: Conduction, Convection and Radiation, Expansion of solids, liquids and gases, application to liquid .
