Surface Area of 10-4 Prisms and Cylinders Warm Up Lesson Presentation Lesson Quiz Holt Geometry

Warm Up Find the perimeter and area of each polygon. 1. a rectangle with base 14 cm and height 9 cm 2. a right triangle with 9 cm and 12 cm legs 3. an equilateral triangle with side length 6 cm P = 46 cm; A = 126 cm2 P = 36 cm; A = 54 cm2

Objectives Learn and apply the formula for the surface area of a prism. Learn and apply the formula for the surface area of a cylinder.

Vocabulary lateral face lateral edge right prism oblique prism altitude surface area lateral surface axis of a cylinder right cylinder oblique cylinder

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.

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. 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.

The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism.

The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh.

The surface area formula is only true for right prisms 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. Caution!

Example 1A: Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of the right rectangular prism. Round to the nearest tenth, if necessary. L = Ph P = 2(9) + 2(7) = 32 ft = 32(14) = 448 ft2 S = Ph + 2B = 448 + 2(7)(9) = 574 ft2

Example 1B: Finding Lateral Areas and Surface Areas of Prisms 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. L = Ph = 30(20) = 600 ft2 P = 3(10) = 30 cm S = Ph + 2B The base area is

Check It Out! Example 1 Find the lateral area and surface area of a cube with edge length 8 cm. L = Ph = 32(8) = 256 cm2 P = 4(8) = 32 cm S = Ph + 2B = 256 + 2(8)(8) = 384 cm2

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.

Example 2A: Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of the right cylinder. Give your answers in terms of . The radius is half the diameter, or 8 ft. L = 2rh = 2(8)(10) = 160 in2 S = L + 2r2 = 160 + 2(8)2 = 288 in2

Check It Out! Example 2 Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius. Step 1 Use the circumference to find the radius. A = r2 Area of a circle 49 = r2 Substitute 49 for A. Divide both sides by  and take the square root. r = 7

Check It Out! Example 2 Continued Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius. Step 2 Use the radius to find the lateral area and surface area. The height is twice the radius, or 14 cm. L = 2rh = 2(7)(14)=196 in2 Lateral area S = L + 2r2 = 196 + 2(7)2 =294 in2 Surface area

Example 3: Finding Surface Areas of Composite Three-Dimensional Figures Find the surface area of the composite figure.

Example 3 Continued The surface area of the rectangular prism is . A right triangular prism is added to the rectangular prism. The surface area of the triangular prism is . Two copies of the rectangular prism base are removed. The area of the base is B = 2(4) = 8 cm2.

Example 3 Continued The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure. S = (rectangular prism surface area) + (triangular prism surface area) – 2(rectangular prism base area) S = 52 + 36 – 2(8) = 72 cm2

Always round at the last step of the problem Always round at the last step of the problem. Use the value of  given by the  key on your calculator. Remember!