Geometry unit 11.2

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UNIT 11.2 SURFACE AREAS OF UNIT 11.2 SURFACE AREAS OF PRISMS AND CYLINDERSPRISMS AND CYLINDERS

Warm UpFind the perimeter and area ofeach 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

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

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

Objectives

lateral facelateral edgeright prismoblique prismaltitudesurface arealateral surfaceaxis of a cylinderright cylinderoblique cylinder

Vocabulary

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 curvedsurfaces of a three-dimensional figure. The lateralarea 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 base of the rectangle is equal to theperimeter of the base of the prism.

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

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 = 32(14) = 448 ft2

S = Ph + 2B = 448 + 2(7)(9) = 574 ft2

P = 2(9) + 2(7) = 32 ft

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

S = Ph + 2BP = 3(10) = 30 cm

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

S = Ph + 2B = 256 + 2(8)(8) = 384 cm2

P = 4(8) = 32 cm

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 π.

L = 2πrh = 2π(8)(10) = 160π in2

The radius is half the diameter, or 8 ft.

S = L + 2πr2 = 160π + 2π(8)2

= 288π in2

Example 2B: Finding Lateral Areas and Surface Areas of Right Cylinders

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.

Step 1 Use the circumference to find the radius.

C = 2πr Circumference of a circle

24π = 2πr Substitute 24π for C.r = 12 Divide both sides by 2π.

Example 2B Continued

Step 2 Use the radius to find the lateral area and surface area. The height is half the radius, or 6 cm.

L = 2πrh = 2π(12)(6) = 144π cm2

S = L + 2πr2 = 144π + 2π(12)2

= 432π in2

Lateral area

Surface area

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.

Find the lateral area and surface area of a cylinder with a base area of 49π and a height that is 2 times the radius.

Step 1 Use the circumference to find the radius.

A = πr2

49π = πr2

Area of a circleSubstitute 49π for A.Divide both sides by π and take the square root.

Step 2 Use the radius to find the lateral area and surface area. The height is twice the radius, or 14 cm.

L = 2πrh = 2π(7)(14)=196π in2

S = L + 2πr2 = 196π + 2π(7)2 =294π in2

Lateral area

Surface area

Find the lateral area and surface area of a cylinder with a base area of 49π and a height that is 2 times the radius.

Check It Out! Example 2 Continued

Example 3: Finding Surface Areas of Composite Three-Dimensional Figures

Find the surface area of the composite figure.

Example 3 Continued

Two copies of the rectangular prism base are removed. The area of the base is B = 2(4) = 8 cm2.

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

The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure.

Example 3 Continued

S = (rectangular prism surface area) + (triangular prism surface area) – 2(rectangular prism base area)

S = 52 + 36 – 2(8) = 72 cm2

Find the surface area of the composite figure. Round to the nearest tenth.

The surface area of the rectangular prism is S =Ph + 2B = 26(5) + 2(36) = 202 cm2.

The surface area of the cylinder is

S =Ph + 2B = 2π(2)(3) + 2π(2)2 = 20π ≈ 62.8 cm2.

The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure.

S = (rectangular surface area) +

(cylinder surface area) – 2(cylinder base area)

S = 202 + 62.8 — 2(π)(22) = 239.7 cm2

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

Remember!

Example 4: Exploring Effects of Changing Dimensions

The edge length of the cube is tripled. Describe the effect on the surface area.

Example 4 Continued

original dimensions: edge length tripled:

Notice than 3456 = 9(384). If the length, width, and height are tripled, the surface area is multiplied by 32, or 9.

S = 6ℓ2

= 6(8)2 = 384 cm2

S = 6ℓ2

= 6(24)2 = 3456 cm2

The height and diameter of the cylinder are

multiplied by . Describe the effect on the

surface area.

original dimensions: height and diameter halved:S = 2π(112) + 2π(11)(14) = 550π cm2

S = 2π(5.52) + 2π(5.5)(7) = 137.5π cm2

Notice than 550 = 4(137.5). If the dimensions are

halved, the surface area is multiplied by

Example 5: Recreation Application

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?

Example 5 Continued

Pup tent:

Tunnel tent:

The tunnel tent requires less nylon.

A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces below. Compare the surface areas. Which will melt faster?

The 5 cm by 5 cm by 1 cm prism has a surface area of 70 cm2, which is greater than the 2 cm by 3 cm by4 cm prism and about the same as the half cylinder. It will melt at about the same rate as the half cylinder.

Lesson Quiz: Part I

Find the lateral area and the surface area of each figure. Round to the nearest tenth, if necessary.

1. a cube with edge length 10 cm

2. a regular hexagonal prism with height 15 in. and base edge length 8 in.

3. a right cylinder with base area 144π cm2 and a height that is the radius

L = 400 cm2 ; S = 600 cm2

L = 720 in2; S ≈ 1052.6 in2

L ≈ 301.6 cm2; S = 1206.4 cm2

Lesson Quiz: Part II

4. A cube has edge length 12 cm. If the edge length of the cube is doubled, what happens to the surface area?

5. Find the surface area of the composite figure.The surface area is multiplied by 4.

S = 3752 m2

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Geometry unit 11.2

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