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**Surface area and Volumes of Prisms**

Lesson 59 Surface area and Volumes of Prisms

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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

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

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**Vocabulary lateral face lateral edge right prism oblique prism**

altitude surface area lateral surface

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**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.

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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.

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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 the perimeter of the base of the prism.

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**The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as**

S = 2ℓw + 2wh + 2ℓh.

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**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!

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**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 = (7)(9) = 574 ft2

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**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 cm2 P = 3(10) = 30 cm S = Ph + 2B The base area is

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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 = (8)(8) = 384 cm2

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

Find the surface area of the composite figure.

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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.

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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 = – 2(8) = 72 cm2

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**Example 4: Exploring Effects of Changing Dimensions**

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

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**Example 4 Continued original dimensions: edge length tripled: S = 6ℓ2**

24 cm original dimensions: edge length tripled: S = 6ℓ2 S = 6ℓ2 = 6(8)2 = 384 cm2 = 6(24)2 = 3456 cm2 Notice than 3456 = 9(384). If the length, width, and height are tripled, the surface area is multiplied by 32, or 9.

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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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Objectives Learn and apply the formula for the volume of a prism.

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Vocabulary volume

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The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. Cavalieri’s principle says that if two three-dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume.

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**Example 1A: Finding Volumes of Prisms**

Find the volume of the prism. Round to the nearest tenth, if necessary. V = ℓwh Volume of a right rectangular prism = (13)(3)(5) = 195 cm3 Substitute 13 for ℓ, 3 for w, and 5 for h.

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**Example 1B: Finding Volumes of Prisms**

Find the volume of a cube with edge length 15 in. Round to the nearest tenth, if necessary. V = s3 Volume of a cube = (15)3 = 3375 in3 Substitute 15 for s.

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**Example 1C: Finding Volumes of Prisms**

Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary. Step 1 Find the apothem a of the base. First draw a right triangle on one base. The measure of the angle with its vertex at the center is

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Example 1C Continued Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary. So the sides are in ratio The leg of the triangle is half the side length, or 4.5 ft. Solve for a. Step 2 Use the value of a to find the base area. P = 6(9) = 54 ft

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Example 1C Continued Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary. Step 3 Use the base area to find the volume.

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Check It Out! Example 1 Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. Volume of a triangular prism

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**Example 2: Recreation Application**

A swimming pool is a rectangular prism. Estimate the volume of water in the pool in gallons when it is completely full (Hint: 1 gallon ≈ ft3). The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.

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Example 2 Continued Step 1 Find the volume of the swimming pool in cubic feet. V = ℓwh = (25)(15)(19) = 3375 ft3 Step 2 Use the conversion factor to estimate the volume in gallons.

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Example 2 Continued Step 3 Use the conversion factor to estimate the weight of the water. 209,804 pounds The swimming pool holds about 25,187 gallons. The water in the swimming pool weighs about 209,804 pounds.

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Check It Out! Example 2 What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. Step 1 Find the volume of the aquarium in cubic feet. V = ℓwh = (120)(60)(16) = 115,200 ft3

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**Check It Out! Example 2 Continued**

What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. Step 2 Use the conversion factor to estimate the volume in gallons.

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**Check It Out! Example 2 Continued**

What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. Step 3 Use the conversion factor to estimate the weight of the water.

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**Check It Out! Example 2 Continued**

What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. The swimming pool holds about 859,701 gallons. The water in the swimming pool weighs about 7,161,313 pounds.

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Check It Out! Example 4 The length, width, and height of the prism are doubled. Describe the effect on the volume. original dimensions: dimensions multiplied by 2: V = ℓwh V = ℓwh = (1.5)(4)(3) = (3)(8)(6) = 18 = 144 Doubling the dimensions increases the volume by 8 times.

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**Example 5: Finding Volumes of Composite Three-Dimensional Figures**

Find the volume of the composite figure. Round to the nearest tenth. The volume of the rectangular prism is: V = ℓwh = (8)(4)(5) = 160 cm3 The base area of the regular triangular prism is: The volume of the regular triangular prism is: The total volume of the figure is the sum of the volumes.

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