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Lesson Menu Five-Minute Check (over Lesson 12–2) CCSS Then/Now New Vocabulary Key Concept: Lateral Area of a Regular Pyramid Example 1:Lateral Area of a Regular Pyramid Key Concept: Surface Area of a Regular Pyramid Example 2:Surface Area of a Square Pyramid Example 3:Surface Area of a Regular Pyramid Key Concept: Lateral and Surface Area of a Cone Example 4:Real-World Example: Lateral Area of a Cone Example 5:Surface Area of a Cone Concept Summary: Lateral and Surface Areas of Solids

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Over Lesson 12–2 5-Minute Check 1 A.65.3 ft 2 B.80.2 ft 2 C.122.5 ft 2 D.130.7 ft 2 Find the lateral area of the cylinder.

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Over Lesson 12–2 5-Minute Check 1 A.65.3 ft 2 B.80.2 ft 2 C.122.5 ft 2 D.130.7 ft 2 Find the lateral area of the cylinder.

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Over Lesson 12–2 5-Minute Check 2 A.150 in 2 B.126 in 2 C.108 in 2 D.96 in 2 Find the lateral area of the prism.

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Over Lesson 12–2 5-Minute Check 2 A.150 in 2 B.126 in 2 C.108 in 2 D.96 in 2 Find the lateral area of the prism.

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Over Lesson 12–2 5-Minute Check 3 A.320 mm 2 B.400 mm 2 C.494.2 mm 2 D.502 mm 2 Find the surface area of the prism.

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Over Lesson 12–2 5-Minute Check 3 A.320 mm 2 B.400 mm 2 C.494.2 mm 2 D.502 mm 2 Find the surface area of the prism.

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Over Lesson 12–2 5-Minute Check 4 A.962.5 cm 2 B.1005.3 cm 2 C.1243.6 cm 2 D.1407.4 cm 2 Find the surface area of the cylinder.

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Over Lesson 12–2 5-Minute Check 4 A.962.5 cm 2 B.1005.3 cm 2 C.1243.6 cm 2 D.1407.4 cm 2 Find the surface area of the cylinder.

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Over Lesson 12–2 5-Minute Check 5 A.426 m 2 B.451 m 2 C.501 m 2 D.526 m 2 The lateral area of a prism is 476 square meters. The area of one of its bases is 25 square meters. What is the surface area of the prism?

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Over Lesson 12–2 5-Minute Check 5 A.426 m 2 B.451 m 2 C.501 m 2 D.526 m 2 The lateral area of a prism is 476 square meters. The area of one of its bases is 25 square meters. What is the surface area of the prism?

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CCSS Content Standards G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Mathematical Practices 1 Make sense of problems and persevere in solving them. 6 Attend to precision.

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Then/Now You found areas of regular polygons. Find lateral areas and surface areas of pyramids. Find lateral areas and surface areas of cones.

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Vocabulary regular pyramid slant height right cone oblique cone

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Concept

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Example 1 Lateral Area of a Regular Pyramid Find the lateral area of the square pyramid. Lateral area of a regular pyramid P = 2.5 ● 4 or 10, ℓ = 5 Answer:

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Example 1 Lateral Area of a Regular Pyramid Find the lateral area of the square pyramid. Lateral area of a regular pyramid P = 2.5 ● 4 or 10, ℓ = 5 Answer:The lateral area is 25 square centimeters.

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Example 1 A.54 in 2 B.64 in 2 C.108 in 2 D.132 in 2 Find the lateral area of the square pyramid.

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Example 1 A.54 in 2 B.64 in 2 C.108 in 2 D.132 in 2 Find the lateral area of the square pyramid.

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Concept

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Example 2 Surface Area of a Square Pyramid Find the surface area of the square pyramid to the nearest tenth. Step 1Find the slant height c 2 = a 2 + b 2 Pythagorean Theorem ℓ 2 = 6 2 + 4 2 a = 6, b = 4, and c = ℓ ℓ= Simplify.

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Example 2 Surface Area of a Square Pyramid Answer: Step 2Find the perimeter and area of the base. P= 4 ● 8 or 32 mA = 8 2 or 64 m 2 Step 3Find the surface area of the pyramid. S= Pℓ + BSurface area of a regular pyramid __ 1 2 1 2 = (32) + 64P = 32, ℓ =, B = 64 ≈ 179.4Use a calculator.

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Example 2 Surface Area of a Square Pyramid Answer:The surface area of the pyramid is about 179.4 square meters. Step 2Find the perimeter and area of the base. P= 4 ● 8 or 32 mA = 8 2 or 64 m 2 Step 3Find the surface area of the pyramid. S= Pℓ + BSurface area of a regular pyramid __ 1 2 1 2 = (32) + 64P = 32, ℓ =, B = 64 ≈ 179.4Use a calculator.

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Example 2 A.96 in 2 B.124.3 in 2 C.138.5 in 2 D.156 in 2 Find the surface area of the square pyramid to the nearest tenth.

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Example 2 A.96 in 2 B.124.3 in 2 C.138.5 in 2 D.156 in 2 Find the surface area of the square pyramid to the nearest tenth.

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Example 3 Surface Area of a Regular Pyramid Step 1Find the perimeter of the base. P = 6 ● 10.4 or 62.4 cm Find the surface area of the regular pyramid. Round to the nearest tenth.

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Example 3 Surface Area of a Regular Pyramid Step 2Find the length of the apothem and the area of the base. A central angle of the hexagon is or 60°, so the angle formed in the triangle is 30°. ______ 360° 6

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Example 3 Surface Area of a Regular Pyramid tan 30° =Write a trigonometric ratio to find the apothem a. a= Solve for a. a≈ 9.0Use a calculator. A= Pa Area of a regular polygon ≈ (62.4)(9.0)Replace P with 62.4 and a width 9.0. ≈ 280.8Multiply. So, the area of the base is approximately 280.8 cm 2.

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Example 3 Surface Area of a Regular Pyramid Answer: Step 3Find the surface area of the pyramid. S= Pℓ + B Surface area of regular pyramid ≈ (62.4)(15) + 280.8P = 62.4, ℓ = 15, and B ≈ 280.8 ≈ 748.8Simplify.

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Example 3 Surface Area of a Regular Pyramid Answer: The surface area of the pyramid is about 748.8 cm 2. Step 3Find the surface area of the pyramid. S= Pℓ + B Surface area of regular pyramid ≈ (62.4)(15) + 280.8P = 62.4, ℓ = 15, and B ≈ 280.8 ≈ 748.8Simplify.

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Example 3 A.198 in 2 B.228.5 in 2 C.255.5 in 2 D.316.3 in 2 Find the surface area of the regular pyramid. Round to the nearest tenth.

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Example 3 A.198 in 2 B.228.5 in 2 C.255.5 in 2 D.316.3 in 2 Find the surface area of the regular pyramid. Round to the nearest tenth.

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Concept

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Example 4 Lateral Area of a Cone ICE CREAM A sugar cone has an altitude of 8 inches and a diameter of inches. Find the lateral area of the sugar cone. If the cone has a diameter of then the radius Use the altitude and the radius to find the slant height with the Pythagorean Theorem.

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Example 4 Lateral Area of a Cone Step 1Find the slant height ℓ. ℓ 2 =8 2 + 1.25 2 Pythagorean Theorem ℓ 2 ≈65.56Simplify. ℓ≈8.1Take the square root of each side. Step 2Find the lateral area L. L= rℓ Lateral area of a cone ≈ (1.25)(8.1)r = 1.25 and ℓ ≈ 8.1 ≈31.8 Answer:

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Example 4 Lateral Area of a Cone Step 1Find the slant height ℓ. ℓ 2 =8 2 + 1.25 2 Pythagorean Theorem ℓ 2 ≈65.56Simplify. ℓ≈8.1Take the square root of each side. Step 2Find the lateral area L. L= rℓ Lateral area of a cone ≈ (1.25)(8.1)r = 1.25 and ℓ ≈ 8.1 ≈31.8 Answer:The lateral area of the sugar cone is about 31.8 square inches.

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Example 4 A.39.7 in. 2 B.43.6 in. 2 C.48.2 in. 2 D.53.1 in. 2 HATS A conical birthday hat has an altitude of 6 inches and a diameter of 4 inches. Find the lateral area of the birthday hat.

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Example 4 A.39.7 in. 2 B.43.6 in. 2 C.48.2 in. 2 D.53.1 in. 2 HATS A conical birthday hat has an altitude of 6 inches and a diameter of 4 inches. Find the lateral area of the birthday hat.

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Example 5 Surface Area of a Cone Find the surface area of the cone. Round to the nearest tenth. Estimate: S ≈ 3 ● 1.5 ● 3 + 3 ● 2 or 19.5 cm 2 S= rℓ + r 2 Surface area of a cone = (1.4)(3.2) + (1.4) 2 r = 1.4 and ℓ = 3.2 ≈20.2 Answer:

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Example 5 Surface Area of a Cone Find the surface area of the cone. Round to the nearest tenth. Estimate: S ≈ 3 ● 1.5 ● 3 + 3 ● 2 or 19.5 cm 2 S= rℓ + r 2 Surface area of a cone = (1.4)(3.2) + (1.4) 2 r = 1.4 and ℓ = 3.2 ≈20.2 Answer:The surface area of the cone is about 20.2 square centimeters. This is close to the estimate, so the answer is reasonable.

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Example 5 A.58.2 cm 2 B.61.3 cm 2 C.63.6 cm 2 D.70.7 cm 2 Find the surface area of the cone. Round to the nearest tenth.

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Example 5 A.58.2 cm 2 B.61.3 cm 2 C.63.6 cm 2 D.70.7 cm 2 Find the surface area of the cone. Round to the nearest tenth.

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Concept

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End of the Lesson

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