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Exploring Properties of Similar Solids: Edges, Surface Area, and Volume Calculations

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This resource provides an engaging overview of geometric shapes and their properties, focusing on determining edges, surface areas, and volumes. Students will explore the characteristics of prisms and pyramids, learning how to identify similarity through the comparison of corresponding linear measures. Detailed examples and guided practice regarding scale factors, volume ratios, and surface area calculations will enhance understanding. Ideal for reinforcing knowledge of geometric relationships and problem-solving skills.

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Exploring Properties of Similar Solids: Edges, Surface Area, and Volume Calculations

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  1. Warm Up • A shape has 5 faces, and 5 vertices how many edges does the shape have? • A sphere has a radius of 7.5, what is its surface area and volume? • What is the surface area and volume of the shape?

  2. Exploring Similar Solids 12.7

  3. What is similar

  4. Tell whether the given right rectangular prism is similar to the right rectangular prism shown at the right. a. b. EXAMPLE 1 Identify similar solids

  5. = = = a.Lengths Heights Widths The prisms are not similar because the ratios of corresponding linear measures are not all equal. ANSWER 2 1 4 2 1 2 2 1 2 4 3 6 4 2 1 2 3 3 8 2 b.Lengths = Widths Heights The prisms are similar because the ratios of corresponding linear measures are all equal. The scale factor is 2:3. ANSWER EXAMPLE 1 Identify similar solids SOLUTION

  6. 1. for Example 1 GUIDED PRACTICE Tell whether the pair of right solids is similar. Explain your reasoning.

  7. = Lengths 12 = Lengths 9 4 4 3 3 ANSWER 16 12 The solids are similar because the ratios of corresponding sides is in the ratio 4:3 for Example 1 GUIDED PRACTICE SOLUTION

  8. Ratio • What is the pattern? • Side: • Surface Area: • Volume:

  9. Formula

  10. The pyramids are similar. Pyramid P has a volume of 1000cubic inches and Pyramid Q has a volume of 216 cubic inches. Find the scale factor of Pyramid P to Pyramid Q. EXAMPLE 3 Find the scale factor

  11. 10 6 1000 a a = b b 216 5 = 3 a3 = b3 The scale factor of Pyramid P to Pyramid Q is 5:3. ANSWER EXAMPLE 3 Find the scale factor SOLUTION Use Theorem 12.13 to find the ratio of the two volumes. Write ratio of volumes. Find cube roots. Simplify.

  12. Cube C has a surface area of 54 square units and Cube D has a surface area of 150 square units. Find the scale factor of C to D. Find the edge length of C, and use the scale factor to find the volume of D. 3. a a b b 3 5 a2 54 = b2 150 3 5 = = for Examples 2, 3, and 4 GUIDED PRACTICE Use Theorem 12.13 to find the ratio of the two properties. SOLUTION Write ratio of volumes. Find square roots. Simplify.

  13. C edge D edge 3 = 5 D edge = 5 Volume of D = 125 square units for Examples 2, 3, and 4 GUIDED PRACTICE Find the edge length. Surface area = 54 square units Single side = 9 units Edge length = 3 units Find volume of D Use Scale Factor.

  14. Homework • Page 850-851 • # 3- 21 odd

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