Geometry 12.5 Areas and Volumes of Similar Solids

Geometry 12.5 Areas and Volumes of Similar Solids

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.
squares
regular pentagons
equilateral
triangles
circles
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
15
10
6
4
6
6
9
10
15
=
=
2
3
2
3
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?
1. 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
4.8
=
4.0
4.8
40
=
48
=
5
10
6
12
=
5
YES
6
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
14
=
5
10
=
1
2
Heights are
not.
3
9
=
1
2
NO
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
4
5
3
8
•
10
6
SCALE FACTOR: 1:2
Base circumference: 6π:12π
Lateral areas: 15π:60π
Volumes: 12π:96π
1:2
1:4
1:8
Exercises
Find the missing information.
4.
5.
6.
7.
8.
2:7
5:6
3:10
scale factor
2:5
ratio of base perimeters
2:5
ratio of heights
2:5
1:3
ratio of lateral areas
4:25
1:9
4 : 49
ratio of total areas
4:25
ratio of volumes
8:125
1:27
8:343
25:36 9:100
125 :
216
27 :
1000
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√27 = 3
3:4
9:16
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
5
Do #11 on your own.
7
Answers: a. 9:49 b. 27:343
Exercises
2
3
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)
2³
=
3³
x
12g
8
=
27
x
12
27x = 96
x ≈3.6
About 3.6 grams
Homework
pg. 511 WE #1-11 all, 13-19 odd
Formula Quiz/Vocab Quiz on Thursday
Chapter 12 Test on Friday