Surface Area of Prisms and Cylinders Warm Up Lesson Presentation Lesson Quiz

Warm-Up 1. Evaluate 2xy + 2yz + 2xz for x = 9, y = 6, and z = 4. ANSWER 228 2. Evaluate 2πx2 + 2πxy for x = 4 and y = 7. Use 3.14 for π . ANSWER 276.32

Warm-Up 3. Find the area of a circle with radius 8 cm. Use 3.14 for π. ANSWER 200.96 cm2 4. Find the area of a right triangle with hypotenuse 13 in. and a side length of 5 in. ANSWER 30 in.2

Warm-Up 5. Find the area of a regular pentagon with apothem 4.13 ft and side length 6 ft. ANSWER 61.95 ft2

Example 1 Find the surface area of a rectangular prism with height 2 centimeters, length 5 centimeters, and width 6 centimeters. SOLUTION STEP 1 Sketch the prism. Imagine unfolding it to make a net.

Example 1 STEP 2 Find the areas of the rectangles that form the faces of the prism. STEP 3 Add the areas of all the faces to find the surface area. The surface area of the prism is S = 2(12) + 2(10) + 2(30) = 104 cm2.

Example 2 Find the surface area and lateral area of the right pentagonal prism. SOLUTION STEP 1 Find the area of a pentagonal base. Use the formula of a regular polygon from page 793. Apothem a = √ 62 –3.5252 ≈ 4.86 Perimeter P = 5(7.05) = 35.25 Area of base B = aP = (0.5)(4.86)(35.25)  85.66

Example 2 STEP 2 Use the formula for the surface area of a right prism. S = 2B + Ph ≈ 2(85.66) + (35.25)(9) ≈ 488.57 The surface area of the right pentagonal prism is about 488.57 square feet. The lateral area is Ph ≈ (35.25)(9) ≈ 317.25 square feet.

Guided Practice 1. Draw a net of a triangular prism. ANSWER

Guided Practice 2. Find the surface area and lateral area of a right rectangular prism with height 7 inches, length 3 inches, and width 4 inches using (a) a net and (b) the formula for the surface area of a right prism. ANSWER a. Left and right faces: 7 · 4 = 28 in.2 Top and bottom faces: 3 · 4 = 12 in.2 Front and back faces: 3 · 7 = 21 in.2 S = 2(28) + 2(12) + 2(21) = 122 in.2 Lateral area = 2(28) + 2(21) = 98 in.2 b. S = 2B + Ph = 2(3 · 4 ) + 14 · 7 = 122 in.2 Lateral area = Ph = 14 · 7 = 98 in.2

Example 3 You are wrapping a stack of 20 compact discs using a shrink wrap. Each disc is cylindrical with height 1.2 millimeters and radius 60 millimeters. What is the minimum amount of shrink wrap needed to cover the stack of 20 discs? COMPACT DISCS

Example 3 SOLUTION The 20 discs are stacked, so the height of the stack will be 20(1.2) = 24 mm. The radius is 60 millimeters. The minimum amount of shrink wrap needed will be equal to the surface area of the stack of discs. S = 2πr2 + 2πrh Surface area of a cylinder. = 2π(60)2 + 2π(60)(24) Substitute known values. ≈ 31,667 Use a calculator. You will need at least 31,667 square millimeters, or about 317 square centimeters of shrink wrap.

Example 4 Find the height of the right cylinder shown, which has a surface area of 157.08 square meters. SOLUTION Substitute known values in the formula for the surface area of a right cylinder and solve for the height h. S = 2πr2 + 2πrh Surface area of a cylinder. 157.08 = 2π(2.5)2 + 2π(2.5)h Substitute known values. 157.08 = 12.5π + 5πh Simplify. 157.08 – 12.5π = 5πh Subtract 12.5π from each side. 117.81 ≈ 5πh Simplify. Use a calculator. 7.5 ≈ h Divide each side by 5π.

Guided Practice 3. Find the surface area and lateral area of a right cylinder with height 18 centimeters and radius 10 centimeters. Round your answer to two decimal places. 1759.29 cm2; 1130.97 cm2 ANSWER 4. Find the radius of a right cylinder with height 5 feet and surface area 208π square feet. 8 ft ANSWER

Lesson Quiz 1. Find the surface area of the solid formed by the net. ANSWER 288 cm2

Lesson Quiz Find the surface of the right prism or right cylinder. 2. ANSWER 2580 m2

Lesson Quiz Find the surface of the right prism or right cylinder. 3. ANSWER 2261.95 yd2

Lesson Quiz 4. A rectangular prism has surface area 1848 ft2. Solve for x. ANSWER 18