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.

squaresequilateraltriangles

circlesregular pentagons

Similar solids have the same shape but not necessarily the same size.

Similar Solids

Like circles, all spheres are similar.

Similar Solids

64

10

Two solids are similar if and only if their bases are similar and their corresponding lengths are proportional.

15

9

6

6

9=

2

3

4

6=

2

3

The bases are similar rectangles because length and width are proportional.

The corresponding heights are also proportional.

10

15=

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 9.

10

12=

5

6

4

4.8=

4.0

4.8=

40

48=

5

6

3. Two right triangular prisms have heights 4 and 6. Their bases are triangles with sides 3, 4, 5, and 6,8,10, respectively.

YESAll squares are similar.

7

14=

5

10=

1

2

Bases are similar.

3

9=

1

2Heights are not.

NO

Bases are proportional, therefore similar. Heights are not. NO

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: 1:2Base circumference: 6π:12π 1:2Lateral areas: 15π:60π 1:4Volumes: 12π:96π 1:8

6

10

•

8

3

4 5

Exercises

Find the missing information.

4. 5. 6. 7. 8.

scale factor 2 : 5

ratio of base perimeters

ratio of heights 1 : 3

ratio of lateral areas 4 : 49

ratio of total areas

ratio of volumes 125 : 216

27 : 1000

2:5

2:5

4:25

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 radii b. the slant heights c. the lateral areas

³√27 = 3 ³√64 = 4

3:4 3:4 9:16

Exercises

10. Two spheres have radii 5 cm and 7 cm. Find the ratio of:a. the areas b. the volumes

5 7

25:49

5²:7²

125:3435³:7³

Do #11 on your own. Answers: a. 9:49 b. 27:343

Exercises

2

3

6

x=

2x = 18

x = 99 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 cm2, 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 = 16π16π cm²4

9

x

36π=

2³

3³

x

12g=

27x = 96

x ≈3.6About 3.6 grams8

27

x

12=

2 3

Homework

pg. 511 WE #1-11 all, 13-19 odd

Formula Quiz/Vocab Quiz on ThursdayChapter 12 Test on Friday