Geometry Surface Area of Pyramids and Cones CONFIDENTIAL

Warm Up 15 in. 10 in. 17 in. 10 cm 15 cm 8 cm Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth. 2 cm 3 cm 1. 2. 3. 1) 490 in2 2) 62.83 cm2 3) 700 cm2 CONFIDENTIAL

Surface Area of Pyramids and Cones The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. regular pyramid nonregular pyramid Slant height Vertices Lateral faces Altitude Bases Next page: CONFIDENTIAL

The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. s l P = 4s CONFIDENTIAL

Lateral and surface Area of a regular pyramid The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl. The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B. l CONFIDENTIAL

Finding Lateral and surface Area of Pyramids Find the lateral area and surface area of each pyramid. a.) a regular square pyramid with base edge length 5 in. and slant height 9in. L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in. S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in 2 Next page: CONFIDENTIAL

b.)Find the lateral area and surface area of each regular pyramid. Step 1 find the base perimeter and apothem. The base perimeter is 6(4) = 24 m. The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 2 Step 2 Find the lateral area. L = ½ Pl Lateral area of a regular pyramid = ½ (24)(7) = 84 m Substitute 24 for P and 7 for l. Step 3 Find the surface area. S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. 2 CONFIDENTIAL

Now you try! 1) Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10ft. 1) lateral area = 90 ft2 surface area = 105 ft2 CONFIDENTIAL

The vertex of a cone is the point opposite the base The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. Slant height right cone oblique cone Vertices Lateral surfaces axis Base Next page: CONFIDENTIAL

The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. CONFIDENTIAL

Lateral and Surface Area of a right cone The lateral area of a right cone with radius r and slant height l is L = rl. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2 r l r l CONFIDENTIAL

Finding Lateral Area and Surface Area of right cones Find the lateral area and surface area of each cone. Give your answers in terms of . A) A right cone with radius 2 m and slant height 3 m. Next page: CONFIDENTIAL

Find the lateral area and surface area. 5 ft l 12 ft B) Step 1 Use the Pythagorean Theorem to find l. l = 5 + 12 = 13 ft 2 Step 2 Find the lateral area and surface area. CONFIDENTIAL

2) Find the lateral area and surface area of the right cone. Now you try! 2) Find the lateral area and surface area of the right cone. 16 ft 6 ft 2) lateral area = 251.43 ft2 surface area = 452.57 ft2 CONFIDENTIAL

Exploring Effects of Changing Dimensions 3 cm 5 cm The radius and slant height of the right cone tripled. Describe the effect on the surface area. Original dimensions: Radius and slant height tripled: CONFIDENTIAL

3) The surface area is multiplied by 4/9. Now you try! 3) The base edge length and slant height of the regular square pyramid are both multiplied by 2/3 . Describe the effect on the surface area. 12 ft 15 ft 3) The surface area is multiplied by 4/9. CONFIDENTIAL

Finding Surface Area of Composite Three-Dimensional Figures 28 cm 90 cm 45 cm Find the surface area of the composite figure. Next page: CONFIDENTIAL

28 cm 90 cm 45 cm CONFIDENTIAL

4) Find the surface area of the composite figure. Now you try! 4) Find the surface area of the composite figure. 2 yd 4) 20.94 yd2 CONFIDENTIAL

Electronics Application Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? 10 in. The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in. The diameter of the cone is 2(7.5) = 15 in. 2 Substitute 10 for l, the slant height of the cone and the radius of the large circle. Solve it CONFIDENTIAL

Now you try! 5) If the radius of the large circle were 12 in., what would be the radius of the cone? 5) 9 in. CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

1) Describe the endpoints of an axis of a cone. Assessment 1) Describe the endpoints of an axis of a cone. 1) The endpoints at the vertex and the center of the base of the cone joins to form the axis of the cone. CONFIDENTIAL

2) Find the lateral area and surface area of each regular pyramid. b. a. 8 cm 12 cm 15 ft 16 ft 2a) lateral area = 288 cm2 surface area = 576 cm2 2b) lateral area = 480 ft2 surface area = 1472 ft2 CONFIDENTIAL

3) Find the lateral area and surface of each right cone 3) Find the lateral area and surface of each right cone. Given your answer in terms of . b. a. 14 m. 22 m. 24 in. 25 in. 3a) lateral area = 175∏ in2 surface area = 224∏ in2 3b) lateral area = 308∏ m2 surface area = 504∏ m2 CONFIDENTIAL

a. The dimensions are cut in half. b. The dimension are tripled. 4) Describe the effect of each change on the surface area of the given figure. a. The dimensions are cut in half. b. The dimension are tripled. 10 in. 6 in. 9 cm. 15 cm. 4a) lateral area = 175∏ in2 surface area = 224∏ in2 4b) lateral area = 308∏ m2 surface area = 504∏ m2 CONFIDENTIAL

5) Find the surface area of each composite figure. b. 15 ft. 8 ft. 18 ft. 15 m. 12 m. 32 m. 26 m. 5a) 1184 ft2 5b) 3498.95 m2 CONFIDENTIAL

6) Anna is making a birthday hat from a pattern that is ¾ of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain. 6 in. 6 ) Anna’s head will fit in the hat as the diameter of the cone will be 9 inches. CONFIDENTIAL

Let’s review CONFIDENTIAL

Surface Area of Pyramids and Cones The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. regular pyramid nonregular pyramid Slant height Vertices Lateral faces Altitude Bases Next page: CONFIDENTIAL

The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. s l P = 4s CONFIDENTIAL

Lateral and surface Area of a regular pyramid The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl. The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B. l CONFIDENTIAL

Finding Lateral and surface Area of Pyramids Find the lateral area and surface area of each pyramid. a.) a regular square pyramid with base edge length 5 in. and slant height 9in. L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in. S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in 2 Next page: CONFIDENTIAL

b.)Find the lateral area and surface area of each regular pyramid. Step 1 find the base perimeter and apothem. The base perimeter is 6(4) = 24 m. The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 2 Step 2 Find the lateral area. L = ½ Pl Lateral area of a regular pyramid = ½ (24)(7) = 84 m Substitute 24 for P and 7 for l. Step 3 Find the surface area. S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. 2 CONFIDENTIAL

The vertex of a cone is the point opposite the base The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. Slant height right cone oblique cone Vertices Lateral surfaces axis Base Next page: CONFIDENTIAL

The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. CONFIDENTIAL

Lateral and Surface Area of a right cone The lateral area of a right cone with radius r and slant height l is L = rl. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2 r l r l CONFIDENTIAL

Finding Lateral Area and Surface Area of right cones Find the lateral area and surface area of each cone. Give your answers in terms of . A) A right cone with radius 2 m and slant height 3 m. Next page: CONFIDENTIAL

Find the lateral area and surface area. 5 ft l 12 ft B) Step 1 Use the Pythagorean Theorem to find l. l = 5 + 12 = 13 ft 2 Step 2 Find the lateral area and surface area. CONFIDENTIAL

Exploring Effects of Changing Dimensions 3 cm 5 cm The radius and slant height of the right cone tripled. Describe the effect on the surface area. Original dimensions: Radius and slant height tripled: CONFIDENTIAL

Finding Surface Area of Composite Three-Dimensional Figures 28 cm 90 cm 45 cm Find the surface area of the composite figure. Next page: CONFIDENTIAL

28 cm 90 cm 45 cm CONFIDENTIAL

Electronics Application Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? 10 in. The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in. The diameter of the cone is 2(7.5) = 15 in. 2 Substitute 10 for l, the slant height of the cone and the radius of the large circle. Solve it CONFIDENTIAL

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