Surface Area and Volume of Similar Figures Unit 5, Lesson 10 Mrs. King

Volume of Similar Figures

What we discovered: If the similarity ratio of two similar solids is a:b, then The ratio of corresponding surface areas is a2:b2 The ratio of corresponding volumes is a3:b3 This applies to all similar solids, not just similar rectangular prisms.

Are the two solids similar? If so, give the similarity ratio. Both solid figures have the same shape. Check that the ratios of the corresponding dimensions are equal. The ratio of the radii is , and the ratio of the height is . 8 26 3 9 The cones are not similar because = . 8 26 3 9 /

Example 1: Find the similarity ratio of two similar cylinders with surface areas of 98π ft2 and 2π ft2. Use the ratio of the surface areas to find the similarity ratio. = a2 b2 98 2 = 49 1 = 7 1 a b

Example 2 The surface are of two similar cylinders are 196 in2 and 324 in2. The volume of the smaller cylinder is 686in3. What is the volume of the larger cylinder? First, find the similarity ratio of the sides! 196 324 REDUCE! 49 81 Find the square root 7 is the similarity ratio of the sides 9

Example 2 The surface are of two similar cylinders are 196 in2 and 324 in2. The volume of the smaller cylinder is 686in3. What is the volume of the larger cylinder? Now, create the similarity ratio of the sides into the similarity ratio of the volumes 73 = 343 93 729 Now, set up a new proportion 686 = 343 x 729 x = 1458

Example 3: The surface area of two similar solids are 160 m2 and 250 m2. The volume of the larger one is 250 m3. What is the volume of the smaller one? V = 128 m3

Example: Two similar square pyramids have volumes of 48 cm3 and 162 cm3. The surface area of the larger pyramid is 135 cm2. Find the surface area of the smaller pyramid. Step 1: Use the ratio of the volumes to find the similarity ratio. = a3 b3 48 162 = 8 27 = 2 3 a b Step 2: Use the similarity ratio to find the surface area of the smaller pyramid. S 135 = 4 9 = 22 32 S = 60

Example A box of detergent shaped like a rectangular prism is 6 in. high and holds 3.25 lb of detergent. How much detergent would a similar box that is 8 in. tall hold? Round your answer to the nearest tenth. The ratio of the heights is 6 : 8, or 3 : 4 in simplest terms. Because the weights are proportional to the volumes, the ratio of the weights equals 33 : 43, or 27 : 64. = 3.25 x 27 64 27x = 208 x 7.7037037 The box that is 8 in. tall would hold about 7.7 lb of detergent.