Geometry 12.5 Areas and Volumes of Similar Solids

Similar polygons have the same shape but not necessarily the same size. Review: Similar Polygons Angles are congruent and sides are proportional.

Please turn to your vocab list and add… -Similar solids -Area ratio of similar solids -Volume ratio of similar solids

Regular polygons and circles are always similar to each other. Review: Similar Polygons The scale factor describes the relationships of the sides or radii. squares equilateral triangles circles regular pentagons

Similar solids have the same shape but not necessarily the same size. Similar Solids Like circles, all spheres are similar.

Similar Solids Two solids are similar if and only if their bases are similar and their corresponding lengths are proportional = = 2 3 The bases are similar rectangles because length and width are proportional. The corresponding heights are also proportional = 2 3 Scale Factor: 2:3

Similar Solids To determine if two solids are similar: (1) Find out if their BASES are similar (3) Check that the heights are to scale. regular polygons are always similar for other polygons, check if sides are proportional (2) Compute the scale factor

Are the given solids similar? 1. Two regular square pyramids have heights 10 and 12. The bases are squares with sides 4 and 4.8, respectively. 2. One rectangular solid has length 7, width 5, and height 3. Another rectangular solid has length 14, width 10, and height = = = = Two right triangular prisms have heights 4 and 6. Their bases are triangles with sides 3, 4, 5, and 6,8,10, respectively. YES All squares are similar = 5 10 = 1 2 Bases are similar. 3 9 = 1 2 Heights are not. NO Bases are proportional, therefore similar. Heights are not.

If the scale factor of two solids is a:b, then (1)the ratio of corresponding perimeters is a:b (2)the ratio of base areas, of lateral areas, and of the total area is a²:b² (3) the ratio of volumes is a³:b³ Scale Factor SCALE FACTOR: 1:2 Base circumference: 6π:12π 1:2 Lateral areas: 15π:60π 1:4 Volumes: 12π:96π 1:

Exercises Find the missing information scale factor2 : 5 ratio of base perimeters ratio of heights1 : 3 ratio of lateral areas4 : 49 ratio of total areas ratio of volumes125 : : :5 4:25 8:125 1:9 1:27 2:7 8:343 5:6 25:36 3:10 9:100

Exercises 9. Two similar cones have volumes 27  and 64 . Find the ratio of: a. the radiib. the slant heightsc. the lateral areas ³√27 = 3³√64 = 4 3:4 9:16

Exercises 10. Two spheres have radii 5 cm and 7 cm. Find the ratio of: a. the areasb. the volumes :49 5²:7² 125:343 5³:7³ Do #11 on your own.Answers: a. 9:49 b. 27:343

Exercises x = 2x = 18 x = 9 9 cm. 12. 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? b. If the area of the larger ball is 36  cm 2, what is the area of the smaller ball? c. If the larger ball weighs 12 g, about how much does the smaller ball weigh? (Hint: Weight is related to volume) 2² 3² x 36π = 9x = 144π x = 16π 16π cm² 4 9 x 36π = 2³ 3³ x 12g = 27x = 96 x ≈3.6 About 3.6 grams 8 27 x 12 = 2 3

Homework pg. 511 WE #1-11 all, odd Formula Quiz/Vocab Quiz on Thursday Chapter 12 Test on Friday