Lesson 9 5 Similar Solids Lesson 9 5

Lesson 9 -5 Similar Solids Lesson 9 -5: Similar Solids 1

Similar Solids Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii) are called similar solids. Lesson 9 -5: Similar Solids 2

Similar Solids Similar solids NOT similar solids Lesson 9 -5: Similar Solids 3

Similar Solids & Corresponding Linear Measures To compare the ratios of corresponding side or other linear lengths, write the ratios as fractions in simplest terms. 6 12 Length: 12 = 3 8 2 3 8 width: 3 2 4 2 height: 6 = 3 4 2 Notice that all ratios for corresponding measures are equal in similar solids. The reduced ratio is called the “scale factor”. Lesson 9 -5: Similar Solids 4

Example: Are these solids similar? Solution: 9 12 6 12 8 16 All corresponding ratios are equal, so the figures are similar Lesson 9 -5: Similar Solids 5

Example: Are these solids similar? Solution: 18 6 8 4 Corresponding ratios are not equal, so the figures are not similar. Lesson 9 -5: Similar Solids 6

Scale Factor and Area What happens to the area when the lengths of the sides of a rectangle are doubled? Ratio of sides = 1: 2 Ratio of areas = 1: 4 What is the scale factor for the two rectangles? The ratio of the areas can be written as Lesson 9 -5: Similar Solids 1: 2 12: 22 7

Similar Solids and Ratios of Areas l l If two similar solids have a scale factor of a : b, then corresponding areas have a ratio of a 2: b 2. This applies to lateral area, surface area, or base area. Ratio of sides = 3: 2 12 8 6 9 9 Surface Area = B + L. A. = 9(9) + (9 + 9 + 9)(12)/2 = 81 +216 = 297 6 Surface Area = B + L. A. = 6(6) + (6 + 6 + 6)(8)/2 = 36 + 96 = 132 Ratio of surface areas: 297: 132 = 9: 4 = 32: 22 8

Scale Factor and Volume What happens to the surface area and volume when the lengths of the sides of a prism are doubled? Ratio of sides = 1: 2 Ratio of areas = 1: 4 Ratio of volumes = 1: 8 The scale factor for the two prisms is The ratio of the surface areas can be written as The ratio of the volumes can be written as Lesson 9 -5: Similar Solids 1: 2 12: 22 13: 23 9

Similar Solids and Ratios of Volumes l If two similar solids have a scale factor of a : b, then their volumes have a ratio of a 3 : b 3. 9 15 6 Ratio of heights = 3: 2 V = r 2 h = (92) (15) = 1215 10 V= r 2 h = (62)(10) = 360 Ratio of volumes: 1215 : 360 = 27: 8 = 33: 23 Lesson 9 -5: Similar Solids 10

Example 1: These two solids are similar. a. The scale factor is b. c. 18: 6 = 3: 1 The ratio of areas is 182: 62 = 32: 12 = 9: 1 The ratio of volumes is 183: 63 = 33: 13 = 27: 1 18 m Lesson 9 -5: Similar Solids 6 m 11

Example 2: These two solids are similar. If the radius of the larger cone is 6 m, what is the radius of the smaller cone? Solution: Write a proportion. 18 m Lesson 9 -5: Similar Solids 6 m 12

Example 3: These two solids are similar. If the lateral area of the smaller cone is 12 , what is the lateral area of the larger cone? Solution: Write a proportion. Use ratio of AREAS. 18 m Lesson 9 -5: Similar Solids 6 m 13

Example 4: Volume of larger is 27 times volume of smaller! These two solids are similar. If the volume of the larger cone is 96 , what is the volume of the smaller cone? Solution: Write a proportion. Use ratio of VOLUMES. 18 m Lesson 9 -5: Similar Solids 6 m 14