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11-1 Study Guide and InterventionSurface Areas of Prisms and Cylinders531719152593Lateral and Surface Areas of Prisms In a solid figure, faces that are not bases are lateral faces. The lateral area is the sum of the area of the lateral faces. The surface area is the sum of the lateral area and the area of the bases.Lateral Area of a PrismIf a prism has a lateral area of L square units, a height of h units, and each base has a perimeter of P units, then L = Ph.Surface Area of a PrismIf a prism has a surface area of S square units, a lateral area of L square units, and each base has an area of B square units, then S = L + 2B or S = Ph + 2B4895850316230Example: Find the lateral and surface area of the regular pentagonal prism above if each base has a perimeter of 75 centimeters and the height is 10 centimeters.L = PhLateral area of a prism= 75(10) P = 75, h = 10= 750 Multiply.S = L + 2B= 750 + 212aP= 750 + 7.5tan 36°(75)≈ 1524.2tan 36° = 7.5aa = 7.5tan 36°The lateral area is 750 square centimeters and the surface area is about 1524.2 square centimeters.ExercisesFind the lateral area and surface area of each prism. Round to the nearest tenth if necessary.345799070651260985704851. 2.260985508003457990672003. 4.325174574253481705571505. 6.11-1 Study Guide and Intervention (continued)Surface Areas of Prisms and Cylinders5177341137944Lateral and Surface Areas of Cylinders A cylinder is a solid with bases that are congruent circles lying in parallel planes. The axis of a cylinder is the segment with endpoints at the centers of these circles. For a right cylinder, the axis is also the altitude of the cylinder.Lateral Area of a CylinderIf a cylinder has a lateral area of L square units, a height of h units, and a base has a radius of r units, then L = 2πrh.Surface Area of a CylinderIf a cylinder has a surface area of S square units, a height of h units, and a base has a radius of r units, then S = L + 2B or 2πrh + 2πr2.Example: Find the lateral and surface area of the cylinder. Round to the nearest tenth.If d = 12 cm, then r = 6 cm.L = 2πrh Lateral area of a cylinder4577379107838= 2π(6)(14) r = 6, h = 14≈ 527.8 Use a calculator.S = 2πrh + 2πr2 Surface area of a cylinder≈ 527.8 + 2π62 2πrh ≈ 527.8, r = 6≈ 754.0 Use a calculator.The lateral area is about 527.8 square centimeters and the surface area is about 754.0 square centimeters.ExercisesFind the lateral area and surface area of each cylinder. Round to the nearest tenth.3450038580332305051098551.2.329299629313437814629313.4.343781491582367030641355.6.11-1 Study Guide and Intervention441198020764500Volumes of Prisms and CylindersVolumes of Prisms The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume is measured in units such as cubic feet, cubic yards, or cubic meters. One cubic unit is the volume of a cube that measures one unit on each edge.Volumeof a PrismIf a prism has a volume of V cubic units, a height of h units, and each base has an area of B square units, then V = Bh.Example 1: Find the volume of the prism.228607048500V = Bh Volume of a prism= (7)(3)(4) B = (7)(3), h = 4= 84 Multiply.The volume of the prism is 84 cubic centimeters.Example 2: Find the volume of the prism if the area of each base is 6.3 square feet.2286014033500V = Bh Volume of a prism= (6.3)(3.5) B = 6.3, h = 3.5= 22.05 Multiply.The volume is 22.05 cubic feet.ExercisesFind the volume of each prism.3903345457200025336553975001.2.2457454889500389509080645003.4.3871595361950024574528575005.6.11-1 Study Guide and Intervention (continued)Volumes of Prisms and Cylinders542480513271500Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. When a solid is not a right solid, use Cavalieri’s Principle to find the volume. The principle states that if two solids have the same height and the same cross sectional area at every level, then they have the same volume.Volume ofa CylinderIf a cylinder has a volume of V cubic units, a height of h units, and the bases have a radius of r units, then V = πr2h.Example 1: Find the volume of the cylinder.2286013843000V = πr2h Volume of a cylinder= π(3)2(4) r = 3, h = 4≈ 113.1 Simplify.The volume is about 113.1 cubic centimeters.Example 2: Find the volume of the oblique cylinder.1587512890500Use the Pythagorean Theorem to find the height of the cylinder.h2 + 52 = 132 Pythagorean Theoremh2 = 144 Simplify.h = 12 Take the positive square root of each side.V = πr2h Volume of a cylinder= π(4)2(12) r = 4, h = 12≈ 603.2 Simplify.The Volume is about 603.2 cubic inches.ExercisesFind the volume of each cylinder. Round to the nearest tenth.3449955527050028511568580001.2.2851155969000341820567310003.4.344995533655002851151905005.6. ................
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