To find the volume of a pyramid and of a cone
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1 - Volumes of Pyramids and Cones Common Core State Standards G-GMD.A. Use volume formulas for... pyramids, cones... to solve problems. G-MG.A. Use geometric sapes, teir measures, and teir properties to describe objects. MP, MP, MP, MP Objective To find te volume of a pyramid and of a cone Look for a pattern in te volumes of te prism and pyramid pairs below. Use te pattern to find te volume of a pyramid wit a base ft by ft and eigt ft. Eplain your reasoning. Make a table and look for a pattern. MATHEMATICAL PRACTICES Pyramid ft volume ft = ft ft in. in. in. Pyramid volume = in. cm cm m m m Pyramid volume = m m cm m Pyramid volume = cm Not to scale m Pyramid volume = m In te Solve It, you analyzed te relationsip between te volume of a prism and te volume of an embedded pyramid. Essential Understanding Te volume of a pyramid is related to te volume of a prism wit te same base and eigt. Teorem - Volume of a Pyramid Te volume of a pyramid is one tird te product of te area of te base and te eigt of te pyramid. V = B B Because of Cavalieri s Principle, te volume formula is true for all pyramids. Te eigt of an oblique pyramid is te lengt of te perpendicular segment from its verte to te plane of te base. Oblique pyramid Capter Surface Area and Volume
2 How is tis similar to finding te volume of a prism? In bot cases, you need te area of te base and te eigt. Problem Finding Volume of a Pyramid Arcitecture Te entrance to te Louvre Museum in Paris, France, is a square pyramid wit a eigt of. m. Wat is te approimate volume of te Louvre Pyramid? Te area of te base of te pyramid is. m #. m, or. m. V = B Use te formula for volume of a pyramid. = (.)(.) Substitute for B and. =. Simplify. Te volume is about m.. m. A sports arena saped like a pyramid as a base area of about, ft and a eigt of ft. Wat is te approimate volume of te arena? How do you use te slant eigt? Te slant eigt is te lengt of te ypotenuse of te rigt triangle. Use te slant eigt to find te eigt of te pyramid. Problem Finding te Volume of a Pyramid Wat is te volume in cubic feet of a square pyramid wit base edges ft and slant eigt ft? Step Find te eigt of te pyramid. c = a + b Use te Pytagorean Teorem. = + Substitute for c, for a, and, or, for b. = + Simplify. = Solve for. = Take te positive square root of bot sides. Step V = B Find te volume of te pyramid. Use te formula for volume of a pyramid. = ( # )() Substitute # for B and for. = Simplify. Te volume of te pyramid is ft.. Wat is te volume of a square pyramid wit base edges m and slant eigt m? ft ft ft ft Lesson - Volumes of Pyramids and Cones
3 Essential Understanding Te volume of a cone is related to te volume of a cylinder wit te same base and eigt. Te cones and te cylinder ave te same base and eigt. It takes tree cones full of rice to fill te cylinder. Teorem - Volume of a Cone Te volume of a cone is one tird te product of te area of te base and te eigt of te cone.. V = B, or V = pr V = B A cone-saped structure can be particularly strong, as downward forces at te verte are distributed to all points in its circular base. B How is tis similar to finding te volume of a cylinder? In bot cases, you need to find te base area of a circle. Problem Finding te Volume of a Cone Traditional Arcitecture Te covering on a tepee rests on poles tat come togeter like concurrent lines. Te resulting structure approimates a cone. If te tepee pictured is ft ig wit a base diameter of ft, wat is its approimate volume? V = pr Use te formula for te volume of a cone. = p() () Substitute, or, for r and for.. Use a calculator. Te volume of te tepee is approimately ft.. a. Te eigt and radius of a cild s tepee are alf tose of te tepee in Problem. Wat is te volume of te cild s tepee to te nearest cubic foot? b. Reasoning Wat is te relationsip between te volume of te original tepee and te cild s tepee? Capter Surface Area and Volume
4 Tis volume formula applies to all cones, including oblique cones. Wat is te eigt of te oblique cone? Te eigt is te lengt of te perpendicular segment from te verte of te cone to te base, wic is ft. In an oblique cone, te segment does not intersect te center of te base. Problem Finding te Volume of an Oblique Cone Wat is te volume of te oblique cone at te rigt? Give your answer in terms of P and also rounded to te nearest cubic foot. V = pr Use te formula for volume of a cone. = p() () Substitute for r and for. = p Simplify.. Use a calculator. Te volume of te cone is p ft, or about ft.. a. Wat is te volume of te oblique cone at te rigt in terms of p and rounded to te nearest cubic meter? b. Reasoning How does te volume of an oblique cone compare to te volume of a rigt cone wit te same diameter and eigt? Eplain. ft ft m m Lesson Ceck Do you know HOW? Wat is te volume of eac figure? If necessary, round to te nearest tent... cm in. in. in. cm Do you UNDERSTAND? MATHEMATICAL PRACTICES. Compare and Contrast How are te formulas for te volume of a pyramid and te volume of a cone alike? How are tey different?. Error Analysis A square pyramid as base edges ft and eigt ft. A cone as diameter ft and eigt ft. Your friend claims te figures ave te same volume because te volume formulas for a pyramid and a cone are te same: V = B. Wat is er error? Practice and Problem-Solving Eercises MATHEMATICAL PRACTICES A Practice Find te volume of eac square pyramid. See Problem.. base edges cm,. base edges in.,. base edges m, eigt cm eigt in. eigt m. Buildings Te Transamerica Pyramid Building in San Francisco is ft tall wit a square base tat is ft on eac side. To te nearest tousand cubic feet, wat is te volume of te Transamerica Pyramid? Lesson - Volumes of Pyramids and Cones
5 Find te volume of eac square pyramid. Round to te nearest tent if necessary.... cm in. m cm in. m m Find te volume of eac square pyramid, given its slant eigt. Round to te nearest tent. See Problem... mm. m ft m mm ft. Cemistry In a cemistry lab you use a filter paper cone to filter a liquid. Te diameter of te cone is. cm and its eigt is cm. How muc liquid will te cone old wen it is full?. Cemistry Tis cone as a filter tat was being used to remove impurities from a solution but became clogged and stopped draining. Te remaining solution is represented by te saded region. How many cubic centimeters of te solution remain in te cone? See Problem. cm cm Find te volume of eac cone in terms of P and also rounded as indicated.. nearest cubic foot. nearest cubic inc. nearest cubic meter See Problem. ft in. m ft in. m B Apply. Tink About a Plan A cone wit radius fits snugly inside a square pyramid, wic fits snugly inside a cube. Wat are te volumes of te tree figures? How can you draw a diagram of te situation? Wat dimensions do te cone, pyramid, and cube ave in common?. Reasoning Suppose te eigt of a pyramid is alved. How does tis affect its volume? Eplain.. Writing Witout doing any calculations, eplain ow te volume of a cylinder wit B = p cm and = cm compares to te volume of a cone wit te same base area and eigt. Capter Surface Area and Volume
6 Find te volume to te nearest wole number..... in. cm ft ft in. Square base cm ft Square base ft Equilateral base. Writing Te two cylinders pictured at te rigt are congruent. How does te volume of te larger cone compare to te total volume of te two smaller cones? Eplain.. Arcitecture Te Pyramid of Peace is an opera ouse in Astana, Kazakstan. Te eigt of te pyramid is approimately m and one side of its square base is approimately m. a. Wat is its volume to te nearest tousand cubic meters? b. How tall would a prism-saped building wit te same square base as te Pyramid of Peace ave to be to ave te same volume as te pyramid?. Hardware Builders use a plumb bob to find a vertical line. Te plumb bob sown at te rigt combines a regular eagonal prism wit a pyramid. Find its volume to te nearest cubic centimeter. cm cm. Reasoning A cone wit radius ft and eigt ft as a volume of p ft. Wat is te volume of te cone formed wen te following appens to te original cone? a. Te radius is doubled. b. Te eigt is doubled. c. Te radius and te eigt are bot doubled. cm Algebra Find te value of te variable in eac figure. Leave answers in simplest radical form. Te diagrams are not to scale.... Volume Volume p r Volume p C Callenge Visualization Suppose you revolve te plane region completely about te given line to sweep out a solid of revolution. Describe te solid. Ten find its volume in terms of P.. te y-ais. te -ais. te line =. te line y = - y O Lesson - Volumes of Pyramids and Cones
7 . A frustum of a cone is te part tat remains wen te top of te cone is cut off by a plane parallel to te base. a. Eplain ow to use te formula for te volume of a cone to find te volume of a frustum of a cone. b. Containers A popcorn container in. tall is te frustum of a cone. Its small radius is. in. and its large radius is in. Wat is its volume?. A disk as radius m. A sector is cut away, and a cone is formed. a. Wat is te circumference of te base of te cone? b. Wat is te area of te base of te cone? c. Wat is te volume of te cone? (Hint: Use te slant eigt and te radius of te base to find te eigt.) H r R Frustum of cone m Standardized Test Prep SAT/ACT. A cone as diameter in. and eigt in. A rectangular prism is in. by in. by in. A square pyramid as base edge in. and eigt in. Wat are te volumes of te tree figures in order from least to greatest? cone, prism, pyramid pyramid, cone, prism prism, cone, pyramid prism, pyramid, cone. One row of a trut table lists p as true and q as false. Wic of te following statements is true? p S q p q p q p. If a polyedron as vertices and edges, ow many faces does it ave? Etended Response. Te point of concurrency of te tree altitudes of a triangle lies outside te triangle. Were are its circumcenter, incenter, and centroid located in relation to te triangle? Draw diagrams to support your answers. Mied Review. A triangular prism as eigt cm. Its base is a rigt triangle wit legs cm and cm. Wat is te volume of te prism?. Given JAC and KIN, you know JA KI, AC IN, and m A m I. Wat can you conclude about JC and KN? See Lesson -. See Lesson -. Get Ready! To prepare for Lesson -, do Eercises and.. Find te area of a circle wit diameter in. to te nearest tent.. Find te circumference of a circle wit radius cm to te nearest centimeter. See Lesson -. Capter Surface Area and Volume
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