Geometry

1
Geometry
12.5 Areas and Volumes of Similar Solids
2
Similar polygons have the same shape but not
necessarily the same size.
Review Similar Polygons
Angles are congruent and sides are proportional.
3

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

4
Regular polygons and circles are always similar
to each other.
Review Similar Polygons
equilateral triangles
squares
circles
regular pentagons
The scale factor describes the relationships of
the sides or radii.
5
Similar solids have the same shape but not
necessarily the same size.
Similar Solids
Like circles, all spheres are similar.
6
Similar Solids
Two solids are similar if and only if their bases
are similar and their corresponding lengths are
proportional.
6
2

Scale Factor 23
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.
7
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.
8
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.
10
5
4
4.0
40
5
YES




12
6
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.
7
5
1
Bases are similar.
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
9
Scale Factor
If the scale factor of two solids is ab,
then (1) the ratio of corresponding perimeters
is ab (2) the ratio of base areas, of lateral
areas, and of the total area
is a²b² (3) the ratio of volumes is a³b³
SCALE FACTOR 12 Base circumference 6p12p
12 Lateral areas 15p60p
14 Volumes 12p96p 18
10
8
5
4

6
3
10
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
27
56
310
25
25
19
2536
9100
425
425
127
8343
8125
11
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
34
34
916
³v27 3
³v64 4
12
Exercises
10. Two spheres have radii 5 cm and 7 cm. Find
the ratio of a. the areas b. the volumes
5²7²
5³7³
2549
125343
7
5
Do 11 on your own.
Answers a. 949 b. 27343
13
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
9x 144p
x
2²
4
16p cm²


x 16p
3²
36p
9
36p
c. If the larger ball weighs 12 g, about how much
does the smaller ball weigh? (Hint Weight
is related to volume)
x
27x 96
x
2³
8
About 3.6 grams


x 3.6
3³
12g
27
12
14
Homework

pg. 511 WE 1-11 all, 13-19 odd Formula
Quiz/Vocab Quiz on Thursday Chapter 12 Test on
Friday