Lesson 9-5: Similar Solids

Lesson 9-5: 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

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

Similar Solids & Corresponding Linear Measures To compare the ratios of corresponding side or other linear lengths, write the ratios as fractions in simplest terms. 12 3 6 8 2 4 Length: 12 = 3 width: 3 height: 6 = 3 8 2 2 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

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

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

Lesson 9-5: Similar Solids 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 1: 2 12: 22 Lesson 9-5: Similar Solids

Similar Solids and Ratios of Areas If two similar solids have a scale factor of a : b, then corresponding areas have a ratio of a2: b2. This applies to lateral area, surface area, or base area. 6 8 Ratio of sides = 3: 2 12 9 9 Surface Area = B + L.A. = 6(6) + (6 + 6 + 6 + 6)(8)/2 = 36 + 96 = 132 Surface Area = B + L.A. = 9(9) + (9 + 9 + 9 + 9)(12)/2 = 81 +216 = 297 Ratio of surface areas: 297:132 = 9:4 = 32: 22

Lesson 9-5: Similar Solids 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 1: 2 12: 22 13: 23 Lesson 9-5: Similar Solids

Similar Solids and Ratios of Volumes If two similar solids have a scale factor of a : b, then their volumes have a ratio of a3 : b3. 9 15 6 10 Ratio of heights = 3:2 V = r2h =  (92) (15) = 1215  V= r2h = (62)(10) = 360  Ratio of volumes: 1215: 360 = 27:8 = 33: 23 Lesson 9-5: Similar Solids

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

Lesson 9-5: Similar Solids 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 6 m Lesson 9-5: Similar Solids

Lesson 9-5: Similar Solids 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 6 m Lesson 9-5: Similar Solids

Lesson 9-5: Similar Solids Volume of larger is 27 times volume of smaller! Example 4: 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 6 m Lesson 9-5: Similar Solids