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Section 12-5 Similar Solids

Section 12-5 Similar Solids. Similar Solids – solids with the same shape but not necessarily the same size. To determine if solids are similar, determine if all corresponding lengths are in the same proportion.

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Section 12-5 Similar Solids

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  1. Section 12-5 Similar Solids

  2. Similar Solids – solids with the same shape but not necessarily the same size. To determine if solids are similar, determine if all corresponding lengths are in the same proportion. In prisms, you can also determine if bases are similar (since they are plane figures). All spheres and cubes are similar.

  3. Right Cylinders Bases are similar because all circles are similar. 6 Heights are in the same proportion. 4 12 Cylinder A 8 Cylinder B Scale Factor = 3:2

  4. Not Similar 10 9 15 12 Lengths are not proportional.

  5. Using these cylinders: 6 4 12 Cylinder A 8 Cylinder B Determine the following values and ratios

  6. Find the ratio of the Base Perimeters, Base Areas, Lateral Areas, Total Areas and Volumes. 12π 8π 3:2 36π 16π 9:4 144π 64π 9:4 216π 96π 9:4 432π 128π 27:8

  7. Theorem 12-11: If the scale factor of two similar solids is a:b, then • The ratio of the corresponding lengths and base perimeters is a:b • The ratio of the areas (base areas, lateral areas, total areas) is a²:b² • The ratio of the volumes is a³:b³

  8. Example 1: Two similar prisms have a scale factor of 4:7. Find the ratio of the lateral area and volume. Lateral Areas = 16:49 Volumes = 64:343 Example 2: The total areas of two spheres are 144 and 4. Find the ratio of their volumes. Total Areas = 144π:4π = 36:1 Scale Factor = 6:1 Ratio of Volumes = 216:1

  9. Example 3: Two similar square pyramids have base areas 4m² and 36m². Find the ratio of the heights of the pyramids. Base Areas = 4:36 = 1:9 Scale Factor = 1:3 Ratio of Heights = 1:3 If the height of the larger pyramid is 27m, what is the height of the smaller pyramid? Height of Smaller Pyramid = 9m

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