12.2 Investigate Surface Area

Transcription

1 Investigating g Geometry ACTIVITY Use before Lesson Investigate Surface Area MATERIALS grap paper scissors tape Q U E S T I O N How can you find te surface area of a polyedron? A net is a pattern tat can be folded to form a polyedron. To find te surface area of a polyedron, you can find te area of its net. E X P L O R E Create a polyedron using a net STEP 1 Draw a net Copy te net below on grap paper. Be sure to label te sections of te net. F B C D E A STEP 2 Create a polyedron Cut out te net and fold it along te black lines to form a polyedron. Tape te edges togeter. Describe te polyedron. Is it regular? Is it convex? STEP 3 Find surface area Te surface area of a polyedron is te sum of te areas of its faces. Find te surface area of te polyedron you just made. (Eac square on te grap paper measures 1 unit by 1 unit.) D R A W C O N C L U S I O N S Use your observations to complete tese exercises 1. Lay te net flat again and find te following measures. A: te area of Rectangle A P: te perimeter of Rectangle A : te eigt of Rectangles B, C, D, and E 2. Use te values from Exercise 1 to find 2A 1 P. Compare tis value to te surface area you found in Step 3 above. Wat do you notice? 3. Make a conjecture about te surface area of a rectangular prism. 4. Use grap paper to draw te net of anoter rectangular prism. Fold te net to make sure tat it forms a rectangular prism. Use your conjecture from Exercise 3 to calculate te surface area of te prism. 802 Capter 12 Surface Area and Volume of Solids

2 12.2 Surface Area of Prisms and Cylinders Before You found areas of polygons. Now You will find te surface areas of prisms and cylinders. Wy? So you can find te surface area of a drum, as in Ex. 22. Key Vocabulary prism lateral faces, lateral edges surface area lateral area net rigt prism oblique prism cylinder rigt cylinder A prism is a polyedron wit two congruent faces, called bases, tat lie in parallel planes. Te oter faces, called lateral faces, are parallelograms formed by connecting te corresponding vertices of te bases. Te segments connecting tese vertices are lateral edges. Prisms are classified by te sapes of teir bases. Te surface area of a polyedron is te sum of te areas of its faces. Te lateral area of a polyedron is te sum of te areas of its lateral faces. Imagine tat you cut some edges of a polyedron and unfold it. Te two-dimensional representation of te faces is called a net. As you saw in te Activity on page 802, te surface area of a prism is equal to te area of its net. base base lateral faces lateral edges E X A M P L E 1 Use te net of a prism Find te surface area of a rectangular prism wit eigt 2 centimeters, lengt 5 centimeters, and widt 6 centimeters. Solution STEP 1 Sketc te prism. Imagine unfolding it to make a net. 2 cm 2 cm 6 cm 5 cm 5 cm 2 cm 5 cm STEP 2 Find te areas of te rectangles tat form te faces of te prism. 6 cm 2 cm Congruent faces Dimensions Area of eac face Left and rigt faces 6 cm by 2 cm 6p cm 2 Front and back faces 5 cm by 2 cm 5p cm 2 Top and bottom faces 6 cm by 5 cm 6p cm 2 STEP 3 Add te areas of all te faces to find te surface area. c Te surface area of te prism is S 5 2(12) 1 2(10) 1 2(30) cm Surface Area of Prisms and Cylinders 803

3 RIGHT PRISMS Te eigt of a prism is te perpendicular distance between its bases. In a rigt prism, eac lateral edge is perpendicular to bot bases. A prism wit lateral edges tat are not perpendicular to te bases is an oblique prism. eigt eigt Rigt rectangular prism THEOREM Oblique triangular prism For Your Notebook THEOREM 12.2 Surface Area of a Rigt Prism Te surface area S of a rigt prism is S 5 2B 1 P 5 ap 1 P, were a is te apotem of te base, B is te area of a base, P is te perimeter of a base, and is te eigt. B P S 5 2B 1 P 5 ap 1 P E X A M P L E 2 Find te surface area of a rigt prism REVIEW APOTHEM For elp wit finding te apotem, see p Find te surface area of te rigt pentagonal prism. Solution STEP 1 Find te perimeter and area of a base of te prism. Eac base is a regular pentagon. Perimeter P 5 5(7.05) Apotem a 5 Ï }} ø 4.86 STEP 2 Use te formula for te surface area tat uses te apotem. S 5 ap 1 P ø (4.86)(35.25) 1 (35.25)(9) ø Surface area of a rigt prism Substitute known values. Simplify. c Te surface area of te rigt pentagonal prism is about square feet. 6 ft ft a 6 ft ft 7.05 ft 6 ft 9 ft GUIDED PRACTICE for Examples 1 and 2 1. Draw a net of a triangular prism. 2. Find te surface area of a rigt rectangular prism wit eigt 7 inces, lengt 3 inces, and widt 4 inces using (a) a net and (b) te formula for te surface area of a rigt prism. 804 Capter 12 Surface Area and Volume of Solids

4 CYLINDERS A cylinder is a solid wit congruent circular bases tat lie in parallel planes. Te eigt of a cylinder is te perpendicular distance between its bases. Te radius of a base is te radius of te cylinder. In a rigt cylinder, te segment joining te centers of te bases is perpendicular to te bases. Te lateral area of a cylinder is te area of its curved surface. It is equal to te product of te circumference and te eigt, or 2πr. Te surface area of a cylinder is equal to te sum of te lateral area and te areas of te two bases. r Base area r A 5πr 2 2πr 2πr Lateral area A 5 2πr base base radius r eigt at classzone.com Base area A 5πr 2 THEOREM For Your Notebook THEOREM 12.3 Surface Area of a Rigt Cylinder Te surface area S of a rigt cylinder is S 5 2B 1 C 5 2πr 2 1 2πr, were B is te area of a base, C is te circumference of a base, r is te radius of a base, and is te eigt. B 5πr 2 C 5 2πr S 5 2B 1 C 5 2pr 2 1 2pr r E X A M P L E 3 Find te surface area of a cylinder COMPACT DISCS You are wrapping a stack of 20 compact discs using a srink wrap. Eac disc is cylindrical wit eigt 1.2 millimeters and radius 60 millimeters. Wat is te minimum amount of srink wrap needed to cover te stack of 20 discs? Solution Te 20 discs are stacked, so te eigt of te stack will be 20(1.2) 5 24 mm. Te radius is 60 millimeters. Te minimum amount of srink wrap needed will be equal to te surface area of te stack of discs. S 5 2πr 2 1 2πr Surface area of a cylinder 5 2π(60) 2 1 2π(60)(24) Substitute known values. ø 31,667 Use a calculator. c You will need at least 31,667 square millimeters, or about 317 square centimeters of srink wrap Surface Area of Prisms and Cylinders 805

5 E X A M P L E 4 Find te eigt of a cylinder Find te eigt of te rigt cylinder sown, wic as a surface area of square meters. Solution Substitute known values in te formula for te surface area of a rigt cylinder and solve for te eigt. 2.5 m S 5 2πr 2 1 2πr Surface area of a cylinder π(2.5) 2 1 2π(2.5) Substitute known values π 1 5π Simplify π 5 5π Subtract 12.5p from eac side ø 5π Simplify. Use a calculator. 7.5 ø Divide eac side by 5p. c Te eigt of te cylinder is about 7.5 meters. GUIDED PRACTICE for Examples 3 and 4 3. Find te surface area of a rigt cylinder wit eigt 18 centimeters and radius 10 centimeters. Round your answer to two decimal places. 4. Find te radius of a rigt cylinder wit eigt 5 feet and surface area 208π square feet EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 7, 9, and 23 5 STANDARDIZED TEST PRACTICE Exs. 2, 17, 24, 25, and 26 EXAMPLE 1 on p. 803 for Exs VOCABULARY Sketc a triangular prism. Identify its bases, lateral faces, and lateral edges. 2. WRITING Explain ow te formula S 5 2B 1 P applies to finding te surface area of bot a rigt prism and a rigt cylinder. USING NETS Find te surface area of te solid formed by te net. Round your answer to two decimal places in cm ft ft 80 ft 10 in. 20 cm 806 Capter 12 Surface Area and Volume of Solids

6 EXAMPLE 2 on p. 804 for Exs. 6 8 SURFACE AREA OF A PRISM Find te surface area of te rigt prism. Round your answer to two decimal places ft ft 3 ft 3 m 9.1 m 3.5 in. 8 m 2 in. EXAMPLE 3 on p. 805 for Exs SURFACE AREA OF A CYLINDER Find te surface area of te rigt cylinder using te given radius r and eigt. Round your answer to two decimal places r in. 5 2 in. r 5 12 mm 5 40 mm r 5 8 in. 5 8 in. 12. ERROR ANALYSIS Describe and correct te error in finding te surface area of te rigt cylinder. S 5 2π(6 2 ) 1 2π(6)(8) 5 2π(36) 1 2π(48) 5 168π ø 528 cm 2 6 cm 8 cm EXAMPLE 4 on p. 806 for Exs ALGEBRA Solve for x given te surface area S of te rigt prism or rigt cylinder. Round your answer to two decimal places. 13. S yd S m S in. 2 7 yd x 15 yd x 8.2 m 8 in. x 17 in. 16. SURFACE AREA OF A PRISM A triangular prism wit a rigt triangular base as leg lengt 9 units and ypotenuse lengt 15 units. Te eigt of te prism is 8 units. Sketc te prism and find its surface area. 17. MULTIPLE CHOICE Te lengt of eac side of a cube is multiplied by 3. Wat is te cange in te surface area of te cube? A Te surface area is 3 times te original surface area. B Te surface area is 6 times te original surface area. C Te surface area is 9 times te original surface area. D Te surface area is 27 times te original surface area. 18. SURFACE AREA OF A CYLINDER Te radius and eigt of a rigt cylinder are eac divided by Ï } 5. Wat is te cange in surface area of te cylinder? 12.2 Surface Area of Prisms and Cylinders 807

7 19. SURFACE AREA OF A PRISM Find te surface area of a rigt exagonal prism wit all edges measuring 10 inces. 20. HEIGHT OF A CYLINDER Find te eigt of a cylinder wit a surface area of 108π square meters. Te radius of te cylinder is twice te eigt. 21. CHALLENGE Te diagonal of a cube is a segment wose endpoints are vertices tat are not on te same face. Find te surface area of a cube wit diagonal lengt 8 units. Round your answer to two decimal places. PROBLEM SOLVING EXAMPLE 3 on p. 805 for Ex BASS DRUM A bass drum as a diameter of 20 inces and a dept of 8 inces. Find te surface area of te drum. 23. GIFT BOX An open gift box is sown at te rigt. Wen te gift box is closed, it as a lengt of 12 inces, a widt of 6 inces, and a eigt of 6 inces. a. Wat is te minimum amount of wrapping paper needed to cover te closed gift box? b. Wy is te area of te net of te box larger tan te amount of paper found in part (a)? c. Wen wrapping te box, wy would you want more paper tan te amount found in part (a)? 6 in. 12 in. 6 in. 24. EXTENDED RESPONSE A rigt cylinder as a radius of 4 feet and eigt of 10 feet. a. Find te surface area of te cylinder. b. Suppose you can eiter double te radius or double te eigt. Wic do you tink will create a greater surface area? c. Ceck your answer in part (b) by calculating te new surface areas. 25. MULTIPLE CHOICE Wic tree-dimensional figure does te net represent? A B C D WORKED-OUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE

8 26. SHORT RESPONSE A company makes two types of recycling bins. One type is a rigt rectangular prism wit lengt 14 inces, widt 12 inces, and eigt 36 inces. Te oter type is a rigt cylinder wit radius 6 inces and eigt 36 inces. Bot types of bins are missing a base, so te bins ave one open end. Wic recycle bin requires more material to make? Explain. 27. MULTI-STEP PROBLEM Consider a cube tat is built using 27 unit cubes as sown at te rigt. a. Find te surface area of te solid formed wen te red unit cubes are removed from te solid sown. b. Find te surface area of te solid formed wen te blue unit cubes are removed from te solid sown. c. Wy are your answers different in parts (a) and (b)? Explain. 28. SURFACE AREA OF A RING Te ring sown is a rigt cylinder of radius r 1 wit a cylindrical ole of r 2. Te ring as eigt. a. Find te surface area of te ring if r 1 is 12 meters, r 2 is 6 meters, and is 8 meters. Round your answer to two decimal places. b. Write a formula tat can be used to find te surface area S of any cylindrical ring were 0 < r 2 < r 1. r 1 r DRAWING SOLIDS A cube wit edges 1 foot long as a cylindrical ole wit diameter 4 inces drilled troug one of its faces. Te ole is drilled perpendicular to te face and goes completely troug to te oter side. Draw te figure and find its surface area. 30. CHALLENGE A cuboctaedron as 6 square faces and 8 equilateral triangle faces, as sown. A cuboctaedron can be made by slicing off te corners of a cube. a. Sketc a net for te cuboctaedron. b. Eac edge of a cuboctaedron as a lengt of 5 millimeters. Find its surface area. MIXED REVIEW Te sum of te measures of te interior angles of a convex polygon is given. Classify te polygon by te number of sides. (p. 507) PREVIEW Prepare for Lesson 12.3 in Exs Find te area of te regular polygon. (p. 762) 35. A D F E 6 B C 36. K L J P H N 12 M 37. P W U V 9 T S R EXTRA PRACTICE for Lesson 12.2, p. 918 ONLINE QUIZ at classzone.com 809