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Surface Area of Pyramids & Cones

Surface Area of Pyramids & Cones. Section 11-3. Objectives. Find the surface area of pyramids Find the surface area of cones. All About Pyramids.

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Surface Area of Pyramids & Cones

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  1. Surface Area of Pyramids & Cones Section 11-3

  2. Objectives • Find the surface area of pyramids • Find the surface area of cones

  3. All About Pyramids • Pyramid - a polyhedron in which one face (the base) can be any polygon and the other faces (lateral faces) are triangles that meet at a common vertex • Named by the shape of the base • Altitude - the perpendicular segment from the vertex to the plane of the base • Height (h) - the length of the altitude

  4. Pyramids Ctd. • Regular pyramid - base is a regular polygon & lateral faces are congruent isosceles triangles • Slant height (l) - length of the altitude of a lateral face

  5. Formulas

  6. You are given = 12 ft and you found that p= 30, so you can find the lateral area. 12 L.A. = pUse the formula for lateral area of a pyramid. 12 = (30)(12)Substitute. Find the surface area of a square pyramid with base edges 7.5 ft and slant height 12 ft. The perimeter p of the square base is 4 X 7.5 ft, or 30 ft. = 180Simplify.

  7. Find the area of the square base. S.A. = L.A. BUse the formula for surface area of a pyramid. = 180  56.25Substitute. (continued) Because the base is a square with side length 7.5 ft, Bs2 7.52 56.25. = 236.25Simplify. The surface area of the square pyramid is 236.25 ft 2.

  8. You try • Find the S.A. of a square pyramid w/ base edges of 5m and slant height of 3m. • 55m2

  9. 1 2 Use the formula L.A. =p to find the lateral area of the pyramid. The altitude of the pyramid, apothem of the base, and altitude of a lateral face form a right triangle, so you can use the Pythagorean Theorem to find the slant height . = 202 + (4 3)2 = 400 + 48 = 448 Find the lateral area of the hexagonal pyramid below. Round your answer to the nearest whole number.

  10. 12 L.A. =pUse the formula for lateral area. 12 = (48)( 448) Substitute. 507.98425 Use a calculator. (continued) The lateral area of the hexagonal pyramid is about 508 m2.

  11. All About Cones • A cone is pointed like a pyramid, but its base is a circle. • Altitude - the perpendicular segment from the vertex to the center of the base. • Height - the length of the altitude • Slant height (l) - distance from the vertex to a point on the edge of the base.

  12. Formulas

  13. = r+r 2Substitute the formulas for L.A. and B. = (5)(13) + (5)2Substitute. = 65 + 25Simplify. = 90 The surface area of the cone is 90 in.2. Find the surface area of the cone in terms of . S.A. = L.A.+ BUse the formula for surface area of a cone.

  14. Your turn • The radius of the base of a cone is 22m. Its slant height is 10m. Find S.A. in terms of . • 704  m2

  15. Use the formula L.A. =r to find the lateral area of the cone. The altitude of the cone, radius of the base, and slant height form a right triangle. Use the Pythagorean Theorem to find the slant height . = 0.52  1.52 = 0.25  2.25 = 2.5 Leandre uses paper cones to cover her plants in the early spring. The diameter of each cone is 1 ft, and its height is 1.5 ft. How much paper is in the cone? Round your answer to the nearest tenth. The cone’s diameter is 1 ft, so its radius r is 0.5 ft.

  16. L.A. =rUse the formula for lateral area of a cone. = (0.5) 2.5 Substitute 0.5 for r and 2.5 for . 2.4836471 Use a calculator. (continued) Find the lateral area. The lateral area of the cone is about 2.5 ft.2

  17. Your turn • Find the L.A. of a cone w/ radius 15 in and ht = 20 in. • 1178 in2

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