Similar Solids 12.7 Geometry

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

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

All corresponding ratios are equal, so the figures are similar Are these solids similar? Example 1: Solution:

Corresponding ratios are not equal, so the figures are not similar. Are these solids similar? Solution: Example 2:

Ex. 3: Identifying Similar Solids Decide whether the two solids are similar. If so, compare the surface areas and volumes of the solids.

Solution: a. The solids are not similar because the ratios of corresponding linear measures are not equal, as shown. 3 = lengths 2 = heights 2 = widths

Solution: b. The solids are similar because the ratios of corresponding linear measures are equal, as shown. 3 = lengths 2 = heights 2 = widths

More... The surface area and volume of the solids are as follows: PrismSurface AreaVolume Smaller S = 2B + Ph= 2(6) + 10(2) = 32V = Bh = 6(2) = 12 Larger S = 2B + Ph= 2(24) + 20(4) = 128V = Bh = 24(4) = 96 The ratio of side lengths is 1:2. The ratio of the surface areas is 32:128, or 1:4. The ratio of the volumes is 12:96, or 1:8.

Similar Solids and Ratios of Areas 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. Surface Area = base + lateral = = Surface Area = base + lateral = = 37 Ratio of sides = 2: 1 Ratio of surface areas: 148:37 = 4:1 = 2 2 : 1 2 7

9 15 Similar Solids and Ratios of Volumes If two similar solids have a scale factor of a : b, then their volumes have a ratio of a 3 : b Ratio of heights = 3:2 V =  r 2 h =  (9 2 ) (15) = 1215V=  r 2 h =  (6 2 )(10) = 360 Ratio of volumes: 1215:360 = 27:8 = 3 3 : 2 3

Ex. 5: Finding the scale factor of similar solids To find the scale factor of the two cubes, find the ratio of the two volumes. a3a3 b3b = a b 8 12 = = 2 3 Write ratio of volumes. Use a calculator to take the cube root. Simplify.  So, the two cubes have a scale factor of 2:3.