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## Geometry

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**Geometry**12.5 Areas and Volumes of Similar Solids**Review: Similar Polygons**Similar polygons have the same shape but not necessarily the same size. 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**Review: Similar Polygons**Regular polygons and circles are always similar to each other. equilateral triangles squares circles regular pentagons The scale factor describes the relationships of the sides or radii.**Similar Solids**Similar solids have the same shape but not necessarily the same size. 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. 6 2 = Scale Factor: 2:3 9 3 4 2 = 15 6 3 10 10 2 6 = 4 15 3 6 9 The bases are similar rectangles because length and width are proportional. The corresponding heights are also proportional.**Similar Solids**To determine if two solids are similar: (1) Find out if their BASES are similar • regular polygons are always similar • for other polygons, check if sides are • proportional (2) Compute the scale factor (3) Check that the heights are to scale.**Are the given solids similar?**• Two regular square pyramids have heights 10 and 12. • The bases are squares with sides 4 and 4.8, respectively. All squares are similar. 4 4.0 40 5 10 5 YES = = = = 6 12 6 4.8 4.8 48 2. One rectangular solid has length 7, width 5, and height 3. Another rectangular solid has length 14, width 10, and height 9. Bases are similar. 7 5 1 3 1 Heights are not. NO = = = 14 10 2 9 2 3. Two right triangular prisms have heights 4 and 6. Their bases are triangles with sides 3, 4, 5, and 6,8,10, respectively. Bases are proportional, therefore similar. Heights are not. NO**Scale Factor**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: 1:2 Base circumference: 6π:12π 1:2 Lateral areas: 15π:60π 1:4 Volumes: 12π:96π 1:8 10 8 5 4 • 3 6**Exercises**Find the missing information. 2:7 5:6 3:10 2:5 2:5 1:9 25:36 9:100 4:25 4:25 1:27 8:343 8:125**Exercises**9. Two similar cones have volumes 27 and 64. Find the ratio of: a. the radii b. the slant heights c. the lateral areas 3:4 3:4 9:16 ³√27 = 3 ³√64 = 4**Exercises**10. Two spheres have radii 5 cm and 7 cm. Find the ratio of: a. the areas b. the volumes 5²:7² 5³:7³ 25:49 125:343 7 5 Do #11 on your own. Answers: a. 9:49 b. 27:343**Exercises**3 2 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? 2 6 2x = 18 = 9 cm. x = 9 3 x b. If the 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 larger ball weighs 12 g, about how much does the smaller ball weigh? (Hint: Weight is related to volume) x x 27x = 96 2³ 8 About 3.6 grams = = x ≈3.6 3³ 12g 27 12**Homework**pg. 511 WE #1-11 all, 13-19 odd Formula Quiz/Vocab Quiz on Thursday Chapter 12 Test on Friday