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10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Warm Up – No Warm up. Better know radians for your test. Take out your homework. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Objectives Learn and apply the formula for the volume of a prism. Learn and apply the formula for the volume of a cylinder. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Vocabulary volume Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. Cavalieri’s principle says that if two three- dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 1 Continued Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 3 Continued Find the volume of a cylinder with base area and a height equal to twice the radius. Give your answers in terms of and rounded to the nearest tenth. Step 3 Use the radius and height to find the volume. = 2662 cm3 8362.9 cm3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 4: Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by . Describe the effect on the volume. radius and height original dimensions: multiplied by : Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 4 Continued The radius and height of the cylinder are multiplied by . Describe the effect on the volume. Notice that . If the radius and height are multiplied by , the volume is multiplied by , or . Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 5 Find the volume of the composite figure. Round to the nearest tenth. Find the side length s of the base: The volume of the The volume of square prism is: the cylinder is: The volume of the composite is the cylinder minus the rectangular prism. Vcylinder — Vsquare prism = 45 — 90 51.4 cm3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Objectives Learn and apply the formula for the volume of a pyramid. Learn and apply the formula for the volume of a cone. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 6: Finding Volumes of Pyramids Find the volume a rectangular pyramid with length 11 m, width 18 m, and height 23 m. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 7 Continued Find the volume of the regular hexagonal pyramid with height equal to the apothem of the base = 1296 ft3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 9 Continued Find the volume of a cone with base circumference 25 in. and a height 2 in. more than twice the radius. = 1406.25 in3 ≈ 4417.9 in3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 10 Continued Find the volume of this cone. 2560 cm3 8042.5 cm3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 11: Exploring Effects of Changing Dimensions The diameter and height of the cone are divided by 3. Describe the effect on the volume. original dimensions: radius and height divided by 3: Notice that . If the radius and height are divided by 3, the volume is divided by 33, or 27. Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Example 12: Finding Volumes of Composite Three- Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The volume of the cylinder is Vcylinder = r2h = (21)2(35)=15,435 cm3. The volume of the lower cone is The volume of the figure is the sum of the volumes. V = 5145 + 15,435 + 5,880 = 26,460 83,126.5 cm3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Lesson Quiz: 1. Find the volume of a cylinder with base area 196 cm2 and a height equal to the diameter V 17,241.1 cm3 2. The edge length of the cube is tripled. Describe the effect on the volume. The volume is multiplied by 27. 3. Find the volume of the composite figure. Round to the nearest tenth. 9160.9 in3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Lesson Quiz: 4. A cone has radius 2 in. and height 7 in. If the radius and height are multiplied by , describe the effect on the volume. The volume is multiplied by . 5. Find the volume of the composite figure. Give your answer in terms of . 10,800 yd3 Holt Geometry 10-6 & 7 Volume of Prisms, Cylinders, Pyramids and Cones Homework • Pg 702 #14, 18, 20-23, 30, 35, 41-44 • Pg 710 #14, 17, 21-27, 33-36, 39, 46-50 On the top of your quiz: How many minutes per day on average do you use a computer/cell phone? What apps/programs/sites do you spend the most time on? Holt Geometry