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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 Lesson Menu
Find the lateral area of the cylinder. A. 65.3 ft2 B. 80.2 ft2 C. 122.5 ft2 D. 130.7 ft2 5-Minute Check 1
Find the lateral area of the cylinder. A. 65.3 ft2 B. 80.2 ft2 C. 122.5 ft2 D. 130.7 ft2 5-Minute Check 1
Find the lateral area of the prism. A. 150 in2 B. 126 in2 C. 108 in2 D. 96 in2 5-Minute Check 2
Find the lateral area of the prism. A. 150 in2 B. 126 in2 C. 108 in2 D. 96 in2 5-Minute Check 2
Find the surface area of the prism. A. 320 mm2 B. 400 mm2 C. 494.2 mm2 D. 502 mm2 5-Minute Check 3
Find the surface area of the prism. A. 320 mm2 B. 400 mm2 C. 494.2 mm2 D. 502 mm2 5-Minute Check 3
Find the surface area of the cylinder. A. 962.5 cm2 B. 1005.3 cm2 C. 1243.6 cm2 D. 1407.4 cm2 5-Minute Check 4
Find the surface area of the cylinder. A. 962.5 cm2 B. 1005.3 cm2 C. 1243.6 cm2 D. 1407.4 cm2 5-Minute Check 4
The lateral area of a prism is 476 square meters 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? A. 426 m2 B. 451 m2 C. 501 m2 D. 526 m2 5-Minute Check 5
The lateral area of a prism is 476 square meters 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? A. 426 m2 B. 451 m2 C. 501 m2 D. 526 m2 5-Minute Check 5
Mathematical Practices 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. CCSS
You found areas of regular polygons. Find lateral areas and surface areas of pyramids. Find lateral areas and surface areas of cones. Then/Now
regular pyramid slant height right cone oblique cone Vocabulary
Concept
Find the lateral area of the square pyramid. 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: Example 1
Find the lateral area of the square pyramid. 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. Example 1
Find the lateral area of the square pyramid. A. 54 in2 B. 64 in2 C. 108 in2 D. 132 in2 Example 1
Find the lateral area of the square pyramid. A. 54 in2 B. 64 in2 C. 108 in2 D. 132 in2 Example 1
Concept
Find the surface area of the square pyramid to the nearest tenth. Surface Area of a Square Pyramid Find the surface area of the square pyramid to the nearest tenth. Step 1 Find the slant height c2 = a2 + b2 Pythagorean Theorem ℓ2 = 62 + 42 a = 6, b = 4, and c = ℓ ℓ = Simplify. Example 2
Step 2 Find the perimeter and area of the base. Surface Area of a Square Pyramid Step 2 Find the perimeter and area of the base. P = 4 ● 8 or 32 m A = 82 or 64 m2 Step 3 Find the surface area of the pyramid. S = Pℓ + B Surface area of a regular pyramid __ 1 2 __ 1 2 = (32) + 64 P = 32, ℓ = , B = 64 ≈ 179.4 Use a calculator. Answer: Example 2
Step 2 Find the perimeter and area of the base. Surface Area of a Square Pyramid Step 2 Find the perimeter and area of the base. P = 4 ● 8 or 32 m A = 82 or 64 m2 Step 3 Find the surface area of the pyramid. S = Pℓ + B Surface area of a regular pyramid __ 1 2 __ 1 2 = (32) + 64 P = 32, ℓ = , B = 64 ≈ 179.4 Use a calculator. Answer: The surface area of the pyramid is about 179.4 square meters. Example 2
Find the surface area of the square pyramid to the nearest tenth. A. 96 in2 B. 124.3 in2 C. 138.5 in2 D. 156 in2 Example 2
Find the surface area of the square pyramid to the nearest tenth. A. 96 in2 B. 124.3 in2 C. 138.5 in2 D. 156 in2 Example 2
Step 1 Find the perimeter of the base. P = 6 ● 10.4 or 62.4 cm Surface Area of a Regular Pyramid Find the surface area of the regular pyramid. Round to the nearest tenth. Step 1 Find the perimeter of the base. P = 6 ● 10.4 or 62.4 cm Example 3
Step 2 Find the length of the apothem and the area of the base. Surface Area of a Regular Pyramid Step 2 Find 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 Example 3
tan 30° = Write a trigonometric ratio to find the apothem a. Surface Area of a Regular Pyramid tan 30° = Write a trigonometric ratio to find the apothem a. a = Solve for a. a ≈ 9.0 Use a calculator. A = Pa Area of a regular polygon ≈ (62.4)(9.0) Replace P with 62.4 and a width 9.0. ≈ 280.8 Multiply. So, the area of the base is approximately 280.8 cm2. Example 3
Step 3 Find the surface area of the pyramid. Surface Area of a Regular Pyramid Step 3 Find the surface area of the pyramid. S = Pℓ + B Surface area of regular pyramid ≈ (62.4)(15) + 280.8 P = 62.4, ℓ = 15, and B ≈ 280.8 ≈ 748.8 Simplify. Answer: Example 3
Step 3 Find the surface area of the pyramid. Surface Area of a Regular Pyramid Step 3 Find the surface area of the pyramid. S = Pℓ + B Surface area of regular pyramid ≈ (62.4)(15) + 280.8 P = 62.4, ℓ = 15, and B ≈ 280.8 ≈ 748.8 Simplify. Answer: The surface area of the pyramid is about 748.8 cm2. Example 3
Find the surface area of the regular pyramid Find the surface area of the regular pyramid. Round to the nearest tenth. A. 198 in2 B. 228.5 in2 C. 255.5 in2 D. 316.3 in2 Example 3
Find the surface area of the regular pyramid Find the surface area of the regular pyramid. Round to the nearest tenth. A. 198 in2 B. 228.5 in2 C. 255.5 in2 D. 316.3 in2 Example 3
Concept
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. Example 4
Step 1 Find the slant height ℓ. Lateral Area of a Cone Step 1 Find the slant height ℓ. ℓ2 = 82 + 1.252 Pythagorean Theorem ℓ2 ≈ 65.56 Simplify. ℓ ≈ 8.1 Take the square root of each side. Step 2 Find 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: Example 4
Step 1 Find the slant height ℓ. Lateral Area of a Cone Step 1 Find the slant height ℓ. ℓ2 = 82 + 1.252 Pythagorean Theorem ℓ2 ≈ 65.56 Simplify. ℓ ≈ 8.1 Take the square root of each side. Step 2 Find 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. Example 4
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. A. 39.7 in.2 B. 43.6 in.2 C. 48.2 in.2 D. 53.1 in.2 Example 4
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. A. 39.7 in.2 B. 43.6 in.2 C. 48.2 in.2 D. 53.1 in.2 Example 4
Find the surface area of the cone. Round to the nearest tenth. 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 cm2 S = rℓ + r2 Surface area of a cone = (1.4)(3.2) + (1.4)2 r = 1.4 and ℓ = 3.2 ≈ 20.2 Answer: Example 5
Find the surface area of the cone. Round to the nearest tenth. 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 cm2 S = rℓ + r2 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. Example 5
Find the surface area of the cone. Round to the nearest tenth. A. 58.2 cm2 B. 61.3 cm2 C. 63.6 cm2 D. 70.7 cm2 Example 5
Find the surface area of the cone. Round to the nearest tenth. A. 58.2 cm2 B. 61.3 cm2 C. 63.6 cm2 D. 70.7 cm2 Example 5
Concept
End of the Lesson