11.7 Notes.notebook
April 22, 2013
11.7 Areas and Volumes of Similar Solids
Similar solids are solids that have the same shape but necessarily the same size.
All spheres are similar!
To decide whether two other solids are similar, determine whether the bases are similar, and corresponding lengths are proportional.
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11.7 Notes.notebook
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Right cylinders
6
4
12
8
I
II
The bases are similar because all circles are similar. The lengths are proportional because 6 = .
12
4 8
So the solids are similar with scale factor 3/2.
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11.7 Notes.notebook
April 22, 2013
Regular square pyramids
8
10
4 5
12
I
6
II
The bases are similar because all squares are similar. The lengths are proportional because 12 8 10
= = .
6 4 5
So the solids are similar with scale factor 2/1.
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11.7 Notes.notebook
April 22, 2013
Cylinders I and II
Pyramids I and II
Scale factor
Base perimeter (I) Base perimeter (II)
L.A. (I) L.A. (II)
Volume (I) Volume (II)
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11.7 Notes.notebook
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Theorem
If the scale factor of two similar solids is a:b, then:
(1) the ratio of corresponding perimeters is a:b,
(2) the ratio of the base areas, of the lateral areas, and of the total areas is a2:b2,
(3) the ratio of the volumes is a3:b3.
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11.7 Notes.notebook
April 22, 2013
For the similar solids show, find the ratios of the (a) base perimeters, (b) lateral areas, and (c) volumes.
6
10
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11.7 Notes.notebook
April 22, 2013
Thm:
If the scale factor of two similar figures is a:b, then
This theorem is the 2­
dimensional version of this theorem.
• the ratio of the perimeters is a:b.
• the ratio of the areas is a2:b2.
Theorem
If the scale factor of two similar solids is a:b, then:
(1) the ratio of corresponding perimeters is a:b,
(2) the ratio of the base areas, of the lateral areas, and of the total areas is a2:b2,
(3) the ratio of the volumes is a3:b3.
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11.7 Notes.notebook
April 22, 2013
Are the two rectangular prisms similar? If so, what is the scale factor of the first figure to the second figure?
1/2
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11.7 Notes.notebook
April 22, 2013
Are the two rectangular prisms similar? If so, what is the scale factor of the first figure to the second figure?
not ~
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11.7 Notes.notebook
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Are the two cylinders similar? If so, what is the scale factor of the first figure to the second figure?
6:5
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11.7 Notes.notebook
April 22, 2013
Are the two rectangular prisms similar? If so, what is the scale factor of the first figure to the second figure?
not ~
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11.7 Notes.notebook
April 22, 2013
Are the two rectangular prisms similar? If so, what is the scale factor of the first figure to the second figure?
2 : 3
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11.7 Notes.notebook
April 22, 2013
The square prisms shown are similar. What is the scale factor of the smaller prism to the larger prism?
9:11
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11.7 Notes.notebook
April 22, 2013
What is the scale factor of two similar prisms with surface areas 144 m2 and 324 m2?
2:3
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11.7 Notes.notebook
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Are any two square prisms similar? Explain.
No the bases are ~ but the heights may not be in proportion.
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11.7 Notes.notebook
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What is the scale factor of the similar rectangular prisms shown?
5:4
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11.7 Notes.notebook
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The lateral areas of two similar paint cans care 1019 cm2 and 425 cm2. The volume of the smaller can is 1157 cm3. What is the volume of the larger can?
4295 cm3
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11.7 Notes.notebook
April 22, 2013
The volumes of two similar solids are 128 m3 and 250 m3. The surface area of the larger solid is 250 m2. What is the surface area of the smaller solid?
160 m2
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11.7 Notes.notebook
April 22, 2013
The volumes of two similar solids are 40 in3 and 135 in3. If the surface area of the smaller solid is 48 in2, what is the surface area of the larger solid?
108 in2
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11.7 Notes.notebook
April 22, 2013
You can compare the capacities and weights of similar objects. The capacity of an object is the amount of fluid the object can hold. The capacities and weights of similar objects made of the same material are proportional to their volumes.
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11.7 Notes.notebook
April 22, 2013
A bottle that is 10 in. high holds 34 oz of milk. The sandwich shop shown is shaped like a milk bottle. To the nearest thousand ounces how much milk could the building hold?
3,760,000 oz
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11.7 Notes.notebook
April 22, 2013
A marble paperweight shaped like a pyramid weighs 0.15 lb. How much does a similarly shaped marble paperweight weigh if each dimensions is three times as large?
4.05 lb
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11.7 Notes.notebook
April 22, 2013
An office supplies store sells paper clips in small and large boxes. A small box holds about 220 paper clips. The large box is formed by doubling the dimensions of the small box. About how many paper clips should fit in the large box?
≈ 1760 paper clips
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