Understanding Ratios of Areas and Volumes in Similar Solids

1. Areas & Volumes of Similar Solids Objective: 1) To find relationships between the ratios of the areas & volumes of similar solids.

2. Similar Solids • Similar Solids – Have the same shape, & all their corresponding dimensions are proportional. • Proportional – Equal Ratios 6in Heights must be proportional! Radii must be proportional! 4in 15in 10in

3. No! Not the same shape. Yes 2x as big or ½ as large. Ex.1: Are the following pairs of solids proportional?? 8ft 2ft 4cm 2ft 4cm 4ft 1ft 1ft 4cm 4cm 4cm 1 1 4 = = 2 2 8

4. Th (10-12) • If side (similarity) ratio is a:b, then • Ratio of their corresponding areas a2:b2. 2) Ratio of their volumes is a3:b3.

5. Ex.2: Surface area ratio • Find the side (similarity) ratio of two similar cylinders with surface areas of 98ft2 & 2ft2. • Write areas as a ratio. • Reduce • √ 98ft2 49ft2 7ft = = 2ft2 1ft2 1ft ** The height of the large cylinder is 7x bigger than the smaller cylinder. ** The radius of the large cylinder is 7x bigger than the smaller cylinder.

6. Ex.3: Volume Ratio Two similar square pyramids have volumes of 48cm3 & 162cm3. The surface area of the larger pyramid is 135cm2. Find the surface of the smaller pyramid. First find the side ratio. Write the volumes as a ratio. Reduce 3√ Set up a surface area ratio 2cm 48cm3 8cm3 3√8ft3 = = = 3cm 162cm3 27cm3 3√27ft3 22 x 4 x x = 60cm2 = = 32 9 135 135

7. What have we learned?? • In order for two solids to be similar they must be • The same shape • Corresponding parts have to be proportional • If the side ratio is a:b, then • Area ratio is a2:b2 • Volume ratio is a3:b3