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Understanding Surface Area of Pyramids and Cones: Formulas and Examples

This guide provides clear definitions and formulas for calculating the surface area of pyramids and cones. A regular pyramid has a base that is a regular polygon, with congruent isosceles triangle faces. The slant height is the distance from the vertex to the base's edge midpoint. We explore examples for finding surface areas using formulas that include area (B) and perimeter (P). Key examples demonstrate how to compute the surface area for both pyramids and cones, ensuring a comprehensive grasp of these geometric shapes.

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Understanding Surface Area of Pyramids and Cones: Formulas and Examples

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  1. Notes 46 Surface Area of Pyramids and Cones

  2. Vocabulary Regular pyramid- a pyramid whose base is a regular polygon, and the faces are congruent isosceles triangles. Slant height of a pyramid- the distance from the vertex to the midpoint of an edge of the base. Slant height of a cone- the distance from the vertex to a point on the edge of the base.

  3. The base of a regular pyramid is a regular polygon, and the faces are congruent isosceles triangles. The diagram shows a square pyramid. The blue dashed line labeledl is the slant height of the pyramid, the distance from the vertex to the midpoint of an edge of the base.

  4. 1 2 S = lw + Pl 1 2 S = (9 • 9) + (36)(10)‏ Additional Example 1A: Finding the Surface Area of a Pyramid Find the surface area of the pyramid. 1 2 S = B + Pl Use the formula. B = lw Substitute. P = 4(9) = 36 S = 81 + 180 Add. S = 261 m2 The surface area is 261 square meters.

  5. 1 2 1 2 S = bh + Pl 1 2 S = (12)(10.38) + (36)(6)‏ Additional Example 1B: Finding the Surface Area of a Pyramid Find the surface area of the pyramid. 1 2 S = B + Pl Use the formula. B= ½bh. 1 2 S = 62.28 + 108 S = 170.28 The surface area is 170.28 m2.

  6. Check It Out: Example 1A Find the surface area ofeachpyramid.

  7. Check It Out: Example 1B Find the surface area of the pyramid

  8. The diagram shows a cone and its net. The blue dashed line is the slant height of the cone, the distance from the vertex to a point on the edge of the base.

  9. Additional Example 2: Finding the Surface Area of a Cone Find the surface area of the cone. Use 3.14 for . S = r2 + rl Use the formula. S ≈ (3.14)(32) + (3.14)(3)(10)‏ Substitute. S ≈ 28.26 + 94.2 Multiply. S ≈ 122.46 Add. The surface area is about 122.46 square centimeters.

  10. Check It Out: Example 2A Find the surface area of the cone. Use 3.14 for .

  11. Check It Out: Example 2B The dimensions of the cone from Exercise 2a quadrupled. Find the surface area of the cone. Use 3.14 for .

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