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Surface Areas of Prisms and Cylinders 609 11-2 11-2 Surface Areas of Prisms 11-2 and Cylinders 1. Plan Objectives What You’ll Learn Check Skills You’ll Need GO for Help Lessons 1-9 and 10-3 1 To find the surface area of a • To find the surface area of a prism Find the area of each net. 40π cm2 363 m2 prism " 2 To find the surface area of a 1.2 2. 3. • To find the surface area of a 96 cm 4 cm cylinder cylinder 4 cm Examples . And Why 8 cm 6 m 1 Finding Surface Area of 4 cm a Prism To find the area covered by a drum on a roller used in road 2 Using Formulas to Find construction, as in Example 4 ␲ Surface Area 4 cm 3 Finding Surface Area of a Cylinder New Vocabulary • prism • bases, lateral faces, altitude, height, lateral area, 4 Real-World Connection surface area (of a prism) • right prism • oblique prism • cylinder • bases, altitude, height, lateral area, surface area (of a cylinder) • right cylinder • oblique cylinder Math Background This lesson uses the area formulas from Chapter 10 and nets to 1 Finding Surface Area of a Prism develop formulas for the lateral and surface areas of prisms and Aprism is a polyhedron with Lateral cylinders. Nets especially simplify exactly two congruent, parallel edges finding these areas for nonright faces, called bases. Other faces prisms, such as parallelepipeds. are lateral faces. You name a The key idea is that the lateral prism by the shape of its bases. Bases Bases faces of any prism are Lateral parallelograms. An altitude of a prism is a perpendicular segment that faces More Math Background: p. 596C joins the planes of the bases. The height h of the prism is Pentagonal prism Triangular prism the length of an altitude. Lesson Planning and A prism may either be right or oblique. Resources Real-World Connection h See p. 596E for a list of the h A triangular prism h resources that support this lesson. breaks white light into rainbow colors. right prisms oblique prism PowerPoint In a right prism the lateral faces are rectangles and a lateral edge is an altitude. Bell Ringer Practice In this book you may assume that a prism is a right prism unless stated or Check Skills You’ll Need pictured otherwise. For intervention, direct students to: Finding Area The lateral area of a prism is the sum of the areas of the lateral faces. The Lesson 1-9: Examples 4–6 surface area is the sum of the lateral area and the area of the two bases. Extra Skills, Word Problems, Proof Practice, Ch. 1 Areas of Regular Polygons 608 Chapter 11 Surface Area and Volume Lesson 10-3: Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 10 Special Needs L1 Below Level L2 Draw examples of prisms on the board and have Making nets of rectangular prisms may help students students identify the bases, lateral faces, and understand and remember Theorem 11-1. altitudes. Point out that in a rectangular prism, any pair of opposite faces could be considered the bases. 608 learning style: visual learning style: tactile 1 EXAMPLE Finding Surface Area of a Prism 2. Teach Use a net to find the surface area of the prism at the left. 4 Surface Area = Lateral Area + area of bases 3 4433 Guided Instruction = sum of areas of lateral faces + area of bases 5 = (5 ? 4 + 5 ? 3 + 5 ? 4 + 5 ? 3) + 2(3)(4) 5 cm 3 Error Prevention! = 70 + 24 4 Some students may think that a 3 cm = 94 right prism must contain a right 4 cm The surface area of the prism is 94 cm2. angle in its base. Point out that the right angle is formed by the Quick Check 1 Use a net to find the surface area lateral faces meeting the base. 5 cm 5 cm of the triangular prism. See left. 12 cm 2 EXAMPLE Alternative Method 1. 216 cm2 6 cm Have students calculate the sum 5 cm You can find formulas for lateral and surface areas by looking at a net for a prism. of the areas of the lateral faces Perimeter of base c to help them understand why the c = a ϩ b ϩ c ϩ d ddBase formula L.A. ph makes sense. 12 cm a b c daa b bcd h PowerPoint hh 6 cm Additional Examples d Base Lateral Area of 1 Use a net to find the surface Perimeter Height c Area a base area of the cube. Lateral Area ϭ ph Surface Area ϭ L.A. ϩ 2B You can use the formulas with any right prism. 2 EXAMPLE Using Formulas to Find Surface Area 11 in. Multiple Choice What is the surface area of the prism? 726 in.2 72 cm2 78 cm2 84 cm2 96 cm2 D E B C 1 A D E B C 2 A D E B C 2 D E Find the surface area of a 3 A B C 4 A D E By the Pythagorean Theorem, the hypotenuse of the B C 3 cm 4 cm 5 A D E B C 10-cm high right prism with Test-Taking Tip triangular base is 5 cm. triangular bases having 18-cm A question could ask L.A. = ph Use the formula for lateral area. edges. Round to the nearest for either surface area 2 = ? p ≠ 3 ± 4 ± 5 ≠ 12 cm whole number. 821 cm or lateral area of a 12 6 6 cm solid. Read the = 72 question carefully. The lateral area of the prism is 72 cm2. Now use the formula for surface area. S.A. = L.A. + 2B = + = ≠ 1 ? ≠ 2 72 2(6) 84 B 2 (3 4) 6 cm The surface area of the prism is 84 cm2. Choice C is the answer. Quick Check 2 Use formulas to find the lateral area and surface area of the prism. 432 m2; about 619 m2 12 m 6 m Lesson 11-2 Surface Areas of Prisms and Cylinders 609 Advanced Learners L4 English Language Learners ELL After Example 2, ask students to write a formula for To help students with the new vocabulary, have them the surface area of a rectangular prism with edges of make and display a poster of a prism and a cylinder, length <, w, and h. S.A. ≠ 2(h< ± hw ± <w) labeling their parts. Clarify that the base of a prism is a face but the base of a polygon is a side. learning style: verbal learning style: visual 609 Guided Instruction Key Concepts Theorem 11-1 Lateral and Surface Areas of a Prism Tactile Learners The lateral area of a right prism is the product of p is the Have students tape the sides of a the perimeter of the base and the height. perimeter sheet of paper together (without h = of a base. overlapping) to form a cylinder. L.A. ph Ask: What is the lateral area of The surface area of a right prism is the sum of B is the area of a base. the cylinder? the area of the the lateral area and the areas of the two bases. paper S.A. = L.A. + 2B Connection to Algebra The formula for the surface area of a cylinder is sometimes written as S.A. = 2pr(r + h). Have 2 students show that this formula 1 Finding Surface Area of a Cylinder is equivalent to the formula S.A. = 2prh + 2pr2. Like a prism, a cylinder has two congruent parallel bases. However, the bases of a cylinder are circles. An altitude of a cylinder is a perpendicular segment that 4 EXAMPLE Diversity joins the planes of the bases. The height h of a cylinder is the length of an altitude. Some students may never h have seen a steamroller. Have other students explain how Bases h h steamrollers work. PowerPoint right cylinders oblique cylinder Additional Examples In this book you may assume that a cylinder is a right cylinder unless stated or pictured otherwise. 3 The radius of the base of a cylinder is 6 ft, and its height is To find the area of the curved surface of a cylinder, visualize “unrolling” it. The 9 ft. Find its surface area in terms area of the resulting rectangle is the lateral area of the cylinder. The surface area of p. 180π ft2 of a cylinder is the sum of the lateral area and the areas of the two circular bases. You can find formulas for these areas by looking at a net for a cylinder. r r Lateral Surface h h Real-World Connection Area Area μ A full turn of the roller inks a 2pr r rectangle with area equal to Area of a base the roller’s lateral area. B ϭ pr2 Key Concepts Theorem 11-2 Lateral and Surface Areas of a Cylinder The lateral area of a right cylinder is the product of the B is the area circumference of the base and the height of the cylinder. of a base. r L.A. = 2prh, or L.A. = pdh h The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. S.A. = L.A. + 2B, or S.A. = 2prh + 2pr2 610 Chapter 11 Surface Area and Volume 610 4 A company sells cornmeal and 3 EXAMPLE Finding Surface Area of a Cylinder barley in cylindrical containers. The diameter of the base of the The radius of the base of a cylinder is 4 in.
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