Solutions of Prisms and Cylinders

Surface Areas of Prisms and Cylinders Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face.   right prism and an oblique prism An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude.   prism with an altitude labeled Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces.   Lateral Area and Surface Area of Right Prisms The lateral area of a right prism with base perimeter P and height h is Ph. The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B. The surface area of a cube with edge length s is S = 6s^2.   The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces.   Example 1: Find the lateral area and surface area of the right rectangular prism. Round to the nearest tenth, if necessary.   right rectangular prism with dimensions 14 ft, 7 ft, and 9 ft LA = P(h) LA = (2(9) + 2(7))(14) LA = (18 + 14)(14) LA = 448 ft^2 SA = L + 2B SA = 448 + 2(126) SA = 448 + 252 SA = 600 ft^2 Example 2: Find the lateral area and surface area of a right regular triangular prism with height 20 cm and base edges of length 10 cm. Round to the nearest tenth, if necessary.   right triangular prism with base edges labeled 10 cm and height labeled 20 cm [LA = Ph] [LA = (30)(20)] [LA = 600 cm^2] [SA = L + 2B] [SA = 600 + 2(\frac{1}{2}(10)(8.66))] [SA = 600 + 86.6] [SA = 686.6 cm^2] The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.   right cylinder and an oblique cylinder Lateral Area and Surface Area of Right Cylinders The lateral area of a right cylinder with radius r and height h is [LA = 2\pi rh] The surface area of a right cylinder with lateral area L and base area B is [SA = L + 2B], or [SA = 2\pi rh + 2\pi r^2] Example 3: Find the lateral area and surface area of the right cylinder. Give your answers in terms of π. right cylinder with height 10 in and radius 8 in [LA = 2\pi rh] [LA = 2\pi(8)(10)] [LA = 160\pi in^2] [SA = L + 2B] [SA = 160\pi + 2\pi(8)^2] [SA = 160\pi + 128\pi] [SA = 288\pi in^2] Example 4: Find the lateral area and surface area of a right cylinder with circumference 24π cm and a height equal to half the radius. Give your answers in terms of π. right cylinder with circumference 24π cm [C = 2\pi r] [24\pi = 2\pi r] [r = 12 cm] [h = \frac{1}{2}r] [h = 6 cm] [LA = 2\pi rh] [LA = 2\pi(12)(6)] [LA = 144\pi cm^2] [SA = L + 2B] [SA = 144\pi + 2\pi(12)^2] [SA = 144\pi + 288\pi] [SA = 432\pi cm^2] Textual Content: Example 5: Find the surface area of the composite figure. composite figure made of a rectangular prism and a triangular prism Rectangular Prism: SA = L + 2B SA = (2)(12) + 2(4)(3) SA = 24 + 24 SA = 48 cm² Triangular Prism: SA = L + 2B SA = (12)(3) + 2(1/2)(4)(3) SA = 36 + 12 SA = 48 cm² Total Surface Area: SA = 48 + 48 - overlap SA = 96 - overlap Example 6: Find the surface area of the composite figure. Round to the nearest tenth. composite figure made of a rectangular prism and a cylinder Rectangular Prism: SA = L + 2B SA = (12)(10) + 2(4)(2) SA = 120 + 16 SA = 136 cm² Cylinder: SA = 2πrh + 2πr² SA = 2π(2)(3) + 2π(2)² SA = 12π + 8π SA = 20π cm² ≈ 62.8 cm² Total Surface Area: SA = 136 + 62.8 - overlap SA ≈ 198.8 cm² Example 7: The edge length of the cube is tripled. Describe the effect on the surface area. cube with edge length 8 cm Original Surface Area: SA = 6s² SA = 6(8)² SA = 384 cm² New Surface Area (after tripling edge length): SA = 6(24)² SA = 3456 cm² Effect: The surface area is multiplied by 9. Example 8: The height and diameter of the cylinder are multiplied by 1/2. Describe the effect on the surface area. cylinder with height 22 cm and diameter 14 cm Original Surface Area: SA = 2πrh + 2πr² SA = 2π(7)(22) + 2π(7)² SA = 308π + 98π SA = 406π cm² ≈ 1274.9 cm² New Surface Area (after multiplying by 1/2): SA = 2π(3.5)(11) + 2π(3.5)² SA = 77π + 24.5π SA = 101.5π cm² ≈ 319.1 cm² Effect: The surface area is multiplied by 1/4. Example 9: A sporting goods company sells tents in two styles, shown below. The sides and floor of each tent are made of nylon. Which tent requires less nylon to manufacture? 1   pup tent and a tunnel tent Pup Tent: SA = 2(1/2)(8)(6) + 2(8)(8) SA = 48 + 128 SA = 176 ft² Tunnel Tent: SA = 2(24)(8) + 2(1/2)(8)(6) SA = 384 + 48 SA = 432 ft²