Surface Areas of Pyramids and Cones LESSON 12–3 Surface Areas of Pyramids and Cones

Five-Minute Check (over Lesson 12–2) TEKS 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 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 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

Mathematical Processes G.1(E), G.1(G) Targeted TEKS G.10(B) Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change. G.11(C) Apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure. Mathematical Processes G.1(E), G.1(G) TEKS

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: 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

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: 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

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: 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

Concept

Step 1 Find the slant height ℓ. Lateral and Surface Area of a Cone Find the lateral area and surface area of the cone. Round to the nearest tenth. Step 1 Find the slant height ℓ. ℓ2 = 1.42 + 3.22 Pythagorean Theorem ℓ2 = 12.2 Simplify. ℓ ≈ 3.5 Take the square root of each side. Example 5

Step 2 Find the lateral area L. Lateral Area of a Cone Step 2 Find the lateral area L. L = rℓ Lateral area of a cone ≈ (1.4)(3.5) r = 1.4 and ℓ ≈ 3.5 ≈ 15.4 Answer: The lateral area of the cone is about 15.4 square centimeters. Example 4

S = rℓ + r2 Surface area of a cone Lateral and Surface Area of a Cone Surface area: 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

Step 1 Find the lateral area of the cone Lateral Area of a Composite Figure Buildings A cylindrical storage building has a bas with a radius of 6 feet. The conical roof is 4 feet tall. The entire building is 12 feet tall. Find the lateral area of the building to the nearest tenth. Step 1 Find the lateral area of the cone ℓ2 = 42 + 62 Pythagorean Theorem ℓ ≈ 7.2 Positive square root. L = Lateral area of a cone ≈ (6)(7.2) Substitute. ≈ 135.7 Simplify  r ℓ Example 5

Lateral Area of a Composite Figure   Answer: 437.3 ft2 Example 5

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

Concept

Surface Areas of Pyramids and Cones LESSON 12–3 Surface Areas of Pyramids and Cones