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Understanding Surface Area Formulas for Pyramids and Cones in Geometry

This lesson focuses on the surface area of pyramids and cones, aligning with the Sunshine State Standards MA.7.G.2.1 and MA.7.G.2.2. Students will learn how to justify and apply formulas for calculating surface areas, using examples to illustrate the process. Key concepts include slant height, base area, and the formulas used for both shapes. Detailed examples are provided for clarity, ensuring students understand how to derive the surface area of different pyramids and cones effectively.

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Understanding Surface Area Formulas for Pyramids and Cones in Geometry

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  1. Lesson Presentation

  2. Sunshine State Standards MA.7.G.2.1 Justify and apply formulas for surface area..of…pyramids…and cones. Also MA.7.G.2.2

  3. Vocabulary regular pyramid slant height of a pyramid slant height of a cone

  4. 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 labeled l is the slant height of the pyramid, the distance from the vertex to the midpoint of an edge of the base.

  5. 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.

  6. 1 2 1 2 S = bh + Pl 1 2 1 2 S = (7)(6) + (21)(8) 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. Substitute. P = 3(7) = 21 S = 21 + 84 Add. S = 105 in2 The surface area is 105 square inches.

  7. 5 m 1 2 S = lw + Pl 3 m 3 m 1 2 S = (3 • 3) + (12)(5) Check It Out: Example 1A Find the surface area of the pyramid. 1 2 S = B + Pl Use the formula. B= lw. Substitute. P = 4(3) = 12 S = 9 + 30 Add. S = 39 m2 The surface area is 39 square meters.

  8. 4 in. 1 2 1 2 S = bh + Pl 1 2 1 2 S = (4)(3) + (12)(6) 6 in. 4 in. 4 in. 3 in. Check It Out: Example 1B Find the surface area of the pyramid 1 2 S = B + Pl Use the formula. B= ½bh. Substitute. P = 3(4) = 12 S = 6 + 36 Add. S = 42 in2 The surface area is 42 square inches.

  9. 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.

  10. 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.

  11. Check It Out: Example 2 Find the surface area of the cone. Use 3.14 for . S = r2 + rl Use the formula. S ≈ (3.14)(42) + (3.14)(4)(10) Substitute. S ≈ 50.24 + 125.6 Multiply. S ≈ 175.84 Add. The surface area is about 175.84 square centimeters.

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