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8.7
How Does The Measure Change?
pg. 20Similar Solids
8.7 – How Does the Measure Change?Similar Solids
As you continue your study of three-dimensional solids, you will explore how the volume of a solid changes as the solid is enlarged proportionally.
8.33 –DIMENSIONSWhat is the difference in dimensions? Examine this idea below.
a. What does an object with zero dimensions look like? Draw a picture. What measurements are found in one dimension? What happens when you double it in size? Triple? Quadruple?
Apoint nothing
b. What does an object with one dimensions look like? Draw a picture. What measurements are found in one dimension? What happens when you double it in size? Triple? Quadruple?
Segment, perimeter x2
A B
x3
x4
c. What does an object with two dimensions look like? Draw a picture. What measurements are found in two dimensions? What happens when you double it in size? Triple? Quadruple?
Plane, polygons
A B
CD
1
1
1original
A B
CD
1
1
A B
CD
2
2
1original
4x2
A B
CD
1
1
A B
CD
3
3
1original
4x2
9x3
A B
CD
1
1
A B
CD
4
4
1original
4x2
9x3
16x4
d. What does an object with three dimensions look like? Draw a picture. What measurements are found in three dimensions? What happens when you double it in size? Triple? Quadruple?
Cubes
1original
1
1
1
1original1
1
1
2
2
2
8x2
1original1
1
1
3
3
3
8x2
27x3
1original1
1
1
4
4
4
8x2
27x3
64x4
8.34 –RATIO OF SIMILAR FIGURESa. Examine the two prisms. Are the two prisms similar?
23
46
=23
812
=23
yes
b. What is the ratio of the edges?
2 323
c. What is the ratio of the Perimeter of the Bases?
12 1823
d. What is the ratio of the Area of the Base?
8 1849
e. What is the ratio of the surface areas?
112 25249
2 8 12 8SA 2 18 18 12SA
f. What is the ratios of the volumes?
64 216 827
8 8V 18 12V
S P A V
r r r2 r3
2 3
g. What if the ratio was different? Complete the table.
Scale Factor
Side Perimeter AreaSurface
AreaVolume
r
6
1
4
75
9
6
1
6
1
36
1
36
1
216
1
4
7
4
7
16
49
16
49
64
3435
9
5
9
25
81
25
81
125
729
r r 2r 2r 3r
8.35 –SIMILAR SOLIDSDetermine if the figures are similar. If they are find the scale factor of the sides, perimeter, area, surface area, and volume.
similar
5/2
5/2
5/2
25/425/4
125/8
Not similar
8.36 –RATIO OF SIMILAR FIGURESThe following figures are similar. Find the scale factor of the solids.
36
121
6
11
S P A V
36
121
2 3
6
11
216
1331
150
2945
7
S P A V25
49
2 3
5
7
125
343
125
216
5
6
S P A V2 3
5
6
25
36
64
125
S P A V64
1254
5
2 3
4
5
16
25
8.37 –RATIO OF SIMILAR FIGURES Corresponding lengths in similar figures are given. Find the ratios (shaded to unshaded) of the perimeters and areas. Find the unknown area.
S A1
162
8
2
1
4 3
x
x 48 ft 2
S A9
259
15
2
3
5
225
x
25x 2025281x in
S A144
3612
6
2
12x 605x in
10
x
S A168
378
2
3
2
2x 2412x cm
8
x
4
9
8.39 –EXTRA PRACTICEa. A shipping box holds 450 golf balls. A similar larger shipping box has dimensions triple the size of the box. How many golf balls does the larger box hold?
S P A V
3 27
450 x 27 = 12,150
2 3
b. Two containers are similar. The smaller container has a surface area of 36m2 and a volume of 8m3. The larger container has a volume of 27m3. Find the larger container's surface area.
2
34
9
S P A V
4
9
8
27
36
x281x m
2 3