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You will be able to calculate the surface area of a pyramid and a cone.

Objectives. You will be able to calculate the surface area of a pyramid and a cone.

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You will be able to calculate the surface area of a pyramid and a cone.

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  1. Objectives You will be able to calculate the surface area of a pyramid and a cone.

  2. 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.

  3. 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.

  4. 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

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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:

  18. 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:

  19. 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.

  20. 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

  21. Write the name of each part of the cone.

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