Ch. 11 Similar Solids

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

Similar Solids Similar solids NOT similar solids

How to determine if solids are similar: Compare the ratios of corresponding sides 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 Ratios for corresponding measures are equal in similar solids. The reduced ratio is called the “scale factor”.

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

Example: Are these solids similar? Solution: 8 18 4 6 Solution: Corresponding ratios are not equal, so the figures are not similar.

Surface Area Ratios Ratio of surface areas: 240:60 = 4:1 = 22: 12 If two similar solids have a scale factor of a : b, then corresponding surface areas have a ratio of a2: b2. 7 5 2 4 3.5 Ratio of sides = 2: 1 8 4 10 Surface Area = ½ Pℓ + B = ½(20)(3.5) + 25 = 60 Surface Area = ½ Pℓ + B = ½(40)(7) + 100 = 240 Ratio of surface areas: 240:60 = 4:1 = 22: 12

Volume Ratios Ratio of volumes: 1215  :360  = 27:8 = 33: 23 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 = Bh =  (92) (15) = 1215 V= Bh = (62)(10) = 360  Ratio of volumes: 1215  :360  = 27:8 = 33: 23

5²:7² 5³:7³ 25:49 125:343 5 7 Exercises Find the ratio of: a. the surface areas b. the volumes 5²:7² 5³:7³ 25:49 125:343 5 7

Two similar cones have volumes 27 and 64. Find the scale factor. Exercises Two similar cones have volumes 27 and 64. Find the scale factor.     ³√27 = 3 ³√64 = 4

Exercises 2 3 Two foam plastic balls have scale factor 2 : 3. a. If the smaller ball has radius 6 cm, what is the radius of the larger ball? 2 6 2x = 18 = 9 cm 3 x x = 9 b. If the surface area of the larger ball is 36 cm2, what is the area of the smaller ball? x x 9x = 144π 2² 4 16π cm² = = x = 16π 3² 36π 9 36π c. If the volume of the smaller ball is 12 cm3, what is the volume of the larger ball? 8x = 324 2³ 12 8 12 40.5 cm3 = = x x x=40.5 3³ 27

Classwork/Homework: Similar Solids WS