1 / 35

12-3

Surface Area of Pyramids and Cones. 12-3. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. 12.3 Surface Area of Pyramids/Cones. Warm Up Find the missing side length of each right triangle with legs a and b and hypotenuse c . 1. a = 7, b = 24 2. c = 15, a = 9

saddam
Download Presentation

12-3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Surface Area of Pyramids and Cones 12-3 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. 12.3 Surface Area of Pyramids/Cones Warm Up Find the missing side length of each right triangle with legs a and b and hypotenuse c. 1.a = 7, b = 24 2.c = 15, a = 9 3.b = 40, c = 41 4.a = 5, b = 5 5.a = 4, c = 8 c = 25 b = 12 a = 9

  3. 12.3 Surface Area of Pyramids/Cones Objectives Learn and apply the formula for the surface area of a pyramid. Learn and apply the formula for the surface area of a cone.

  4. 12.3 Surface Area of Pyramids/Cones Vocabulary vertex of a pyramid regular pyramid slant height of a regular pyramid altitude of a pyramid vertex of a cone axis of a cone right cone oblique cone slant height of a right cone altitude of a cone

  5. 12.3 Surface Area of Pyramids/Cones The vertex of a pyramidis the point opposite the base of the pyramid. The base of a regular pyramidis a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramidis the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramidis the perpendicular segment from the vertex to the plane of the base.

  6. 12.3 Surface Area of Pyramids/Cones 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.

  7. 12.3 Surface Area of Pyramids/Cones

  8. 12.3 Surface Area of Pyramids/Cones Example 1A: Finding Lateral Area and Surface Area of Pyramids Find the lateral area and surface area of a regular square pyramid with base edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if necessary. Lateral area of a regular pyramid P = 4(14) = 56 cm Surface area of a regular pyramid B = 142 = 196 cm2

  9. 12.3 Surface Area of Pyramids/Cones The apothem is , so the base area is Example 1B: Finding Lateral Area and Surface Area of Pyramids Find the lateral area and surface area of the regular pyramid. Step 1 Find the base perimeter and apothem. The base perimeter is 6(10) = 60 in.

  10. 12.3 Surface Area of Pyramids/Cones Example 1B Continued Find the lateral area and surface area of the regular pyramid. Step 2 Find the lateral area. Lateral area of a regular pyramid Substitute 60 for P and 16 for ℓ.

  11. 12.3 Surface Area of Pyramids/Cones Example 1B Continued Find the lateral area and surface area of the regular pyramid. Step 3 Find the surface area. Surface area of a regular pyramid Substitute for B.

  12. 12.3 Surface Area of Pyramids/Cones The apothem is so the base area is Check It Out! Example 1 Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. Step 1 Find the base perimeter and apothem. The base perimeter is 3(6) = 18 ft.

  13. 12.3 Surface Area of Pyramids/Cones Check It Out! Example 1 Continued Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. Step 2 Find the lateral area. Lateral area of a regular pyramid Substitute 18 for P and 10 for ℓ.

  14. 12.3 Surface Area of Pyramids/Cones Substitute for B. Check It Out! Example 1 Continued Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. Step 3 Find the surface area. Surface area of a regular pyramid

  15. 12.3 Surface Area of Pyramids/Cones The vertex of a coneis the point opposite the base. The axis of a coneis the segment with endpoints at the vertex and the center of the base. The axis of a right coneis perpendicular to the base. The axis of an oblique coneis not perpendicular to the base.

  16. 12.3 Surface Area of Pyramids/Cones The slant height of a right coneis the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a coneis a perpendicular segment from the vertex of the cone to the plane of the base.

  17. 12.3 Surface Area of Pyramids/Cones Example 2A: Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of a right cone with radius 9 cm and slant height 5 cm. L = rℓ Lateral area of a cone = (9)(5) = 45 cm2 Substitute 9 for r and 5 for ℓ. S = rℓ + r2 Surface area of a cone = 45 + (9)2 = 126 cm2 Substitute 5 for ℓ and 9 for r.

  18. 12.3 Surface Area of Pyramids/Cones Example 2B: Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of the cone. Use the Pythagorean Theorem to find ℓ. L = rℓ Lateral area of a right cone = (8)(17) = 136 in2 Substitute 8 for r and 17 for ℓ. S = rℓ+ r2 Surface area of a cone = 136 + (8)2 = 200 in2 Substitute 8 for r and 17 for ℓ.

  19. 12.3 Surface Area of Pyramids/Cones ℓ Check It Out! Example 2 Find the lateral area and surface area of the right cone. Use the Pythagorean Theorem to find ℓ. L = rℓ Lateral area of a right cone = (8)(10) = 80 cm2 Substitute 8 for r and 10 for ℓ. S = rℓ + r2 Surface area of a cone = 80 + (8)2 = 144 cm2 Substitute 8 for r and 10 for ℓ.

  20. 12.3 Surface Area of Pyramids/Cones Example 3: Exploring Effects of Changing Dimensions The base edge length and slant height of the regular hexagonal pyramid are both divided by 5. Describe the effect on the surface area.

  21. 12.3 Surface Area of Pyramids/Cones 3 in. 12 12 2 in. S = Pℓ + B S = Pℓ + B Example 3 Continued base edge length and slant height divided by 5: original dimensions:

  22. 12.3 Surface Area of Pyramids/Cones 3 in. 2 in. Notice that . If the base edge length and slant height are divided by 5, the surface area is divided by 52, or 25. Example 3 Continued base edge length and slant height divided by 5: original dimensions:

  23. 12.3 Surface Area of Pyramids/Cones Check It Out! Example 3 The base edge length and slant height of the regular square pyramid are both multiplied by . Describe the effect on the surface area.

  24. 12.3 Surface Area of Pyramids/Cones 8 ft 12 12 10 ft S = Pℓ + B S = Pℓ + B By multiplying the dimensions by two-thirds, the surface area was multiplied by . Check It Out! Example 3 Continued 8 ft original dimensions: multiplied by two-thirds: = 585 cm2 = 260 cm2

  25. 12.3 Surface Area of Pyramids/Cones Example 4: Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. Left-hand cone: The lateral area of the cone is L = rl = (6)(12) = 72 in2. Right-hand cone: Using the Pythagorean Theorem, l = 10 in. The lateral area of the cone is L = rl = (6)(10) = 60 in2.

  26. 12.3 Surface Area of Pyramids/Cones Example 4 Continued Find the surface area of the composite figure. Composite figure: S = (left cone lateral area) + (right cone lateral area) = 60 in2 + 72 in2 = 132 in2

  27. 12.3 Surface Area of Pyramids/Cones Check It Out! Example 4 Find the surface area of the composite figure. Surface Area of Cube without the top side: S = 4wh + B S = 4(2)(2) + (2)(2) = 20 yd2

  28. 12.3 Surface Area of Pyramids/Cones Check It Out! Example 4 Continued Surface Area of Pyramid without base: Surface Area of Composite: Surface of Composite = SA of Cube + SA of Pyramid

  29. 12.3 Surface Area of Pyramids/Cones The area of the pattern is the lateral area of the cone. The area of the pattern is also of the area of the large circle, so Example 5: Manufacturing Application If the pattern shown is used to make a paper cup, what is the diameter of the cup? The radius of the large circle used to create the pattern is the slant height of the cone.

  30. 12.3 Surface Area of Pyramids/Cones Example 5 Continued If the pattern shown is used to make a paper cup, what is the diameter of the cup? Substitute 4 for ℓ, the slant height of the cone and the radius of the large circle. r = 2 in. Solve for r. The diameter of the cone is 2(2) = 4 in.

  31. 12.3 Surface Area of Pyramids/Cones The area of the pattern is the lateral area of the cone. The area of the pattern is also of the area of the large circle, so Check It Out! Example 5 What if…? If the radius of the large circle were 12 in., what would be the radius of the cone? The radius of the large circle used to create the pattern is the slant height of the cone.

  32. 12.3 Surface Area of Pyramids/Cones Check It Out! Example 5 Continued What if…? If the radius of the large circle were 12 in., what would be the radius of the cone? Substitute 12 for ℓ, the slant height of the cone and the radius of the large circle. r = 9 in. Solve for r. The radius of the cone is 9 in.

  33. 12.3 Surface Area of Pyramids/Cones Lesson Quiz: Part I Find the lateral area and surface area of each figure. Round to the nearest tenth, if necessary. 1. a regular square pyramid with base edge length 9 ft and slant height 12 ft 2. a regular triangular pyramid with base edge length 12 cm and slant height 10 cm L = 216 ft2; S = 297 ft2 L = 180 cm2; S ≈ 242.4 cm2

  34. 12.3 Surface Area of Pyramids/Cones The surface area is multiplied by . Lesson Quiz: Part II 4. A right cone has radius 3 and slant height 5. The radius and slant height are both multiplied by . Describe the effect on the surface area. 5. Find the surface area of the composite figure. Give your answer in terms of . S = 24ft2

  35. 12.1-3 Exploring Solids Videos:Surface Area https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-geometry-topic/cc-6th-volume-surface-area/v/nets-of-polyhedra

More Related