Learn to find the volume and surface area of similar three-dimensional figures.

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Learn to find the volume and surface area of
similar three-dimensional figures.
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Recall that similar figures have proportional
side lengths. The surface areas of similar
three-dimensional figures are also proportional.
To see this relationship, you can compare the
areas of corresponding faces of similar
rectangular prisms.
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Area of front of smaller prism
Area of front of larger prism
l w
l w
6 10
3 5
(3 2) (5 2)
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Each dimension has a scale factor of 2.
(3 5) (2 2)
15 22
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The area of the front face of the larger prism is
22 times the area of the front face of the
smaller prism. This is true for the entire
surface area of the prisms.
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Additional Example 1A Finding the Surface Area
of a Similar Figure
The surface area of a box is 35 in2. What is the
surface area of a similar box that is larger by a
scale factor of 7?
Multiply by the square of the scale factor.
S 35 72
S 35 49
Evaluate the power.
S 1,715
Multiply.
The surface area of the larger box is 1,715 in2.
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Additional Example 1B Finding the Surface Area
of a Similar Figure
The surface area of a box is 1,300 in2. Find the
surface area of a similar box that is smaller by
a scale factor of .
1 2
Multiply by the square of the scale factor.
1 2
2
S 1,300
1 4
S 1,300
Evaluate the power.
S 325
Multiply.
The surface area of the smaller box is 325 in2.
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Check It Out Example 1A
The surface area of a box is 50 in2. What is the
surface area of a similar box that is larger by a
scale factor of 3?
Multiply by the square of the scale factor.
S 50 32
Evaluate the power.
S 50 9
Multiply.
S 450
The surface area of the larger box is 450 in2.
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Check It Out Example 1B
The surface area of a box is 1,800 in2. Find the
surface area of a similar box that is smaller by
a scale factor of .
1 3
Multiply by the square of the scale factor.
1 3
2
S 1,800
1 9
S 1,800
Evaluate the power.
S 200
Multiply.
The surface area of the smaller box is 200 in2.
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The volumes of similar three-dimensional figures
are also related.
2 ft
4 ft
1 ft
2 ft
3 ft
6 ft
Volume of smaller box
Volume of larger box
2 3 1
4 6 2
(2 2) (3 2) (1 2)
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(2 3 1) (2 2 2)?
The volume of the larger box is 23 times the
volume of the smaller box.
6 23
Each dimension has a scale factor of 2.
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Additional Example 2 Finding Volume Using
Similar Figures
The volume of a childs swimming pool is 28
ft3. What is the volume of a similar pool that is
larger by a scale factor of 4?
Multiply by the cube of the scale factor.
V 28 43
Evaluate the power.
V 28 64
V 1,792 ft3
Multiply.
Estimate
V 30 60
Round the measurements.
1,800
The answer is reasonable.
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Check It Out Example 2
The volume of a small hot tube is 48 ft3. What is
the volume of a similar hot tub that is larger by
a scale factor of 2?
Use the volume of the smaller prism and the cube
of the scale factor.
V 48 23
Evaluate the power.
V 48 8
V 384 ft3
Multiply.
Estimate
V 50 8
Round the measurements.
400
The answer is reasonable.
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Additional Example 3 Problem Solving Application
The sink in Kevins workshop measures 16 in. by
15 in. by 6 in. Another sink with a similar shape
is larger by a scale factor of 2. There are 231
in3 in 1 gallon. Estimate how many more gallons
the larger sink holds.
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Additional Example 3 Continued
Rewrite the question as a statement.
Compare the capacities of two similar sinks,
and estimate how much more water the larger sink
holds.
List the important information
The smaller sink is 16 in. x 15 in. x 6 in.
The large sink is similar to the small sink by
a scale factor of 2.
231 in3 1 gal
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Additional Example 3 Continued
You can write an equation that relates the volume
of the large sink to the volume of the small
sink. Then convert cubic inches to gallons to
compare the capacities of the sinks.
Volume of large sink
Volume of small sink (a scale factor)3
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Additional Example 3 Continued
Volume of small sink 16 x 15 x 6 1,440 in3
Volume of large sink 1,440 x 23 11,520 in3
Convert each volume into gallons
Subtract the capacities 50 gal 6 gal 44 gal
The large sink holds about 44 gallons more than
the small sink.
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Additional Example 3 Continued
Look Back
Double the dimensions of the small sink and find
the volume 32 x 30 x 12 11,520 in3. Subtract
the volumes of the two sinks 11,520 1,440
10,080 in3. Convert this measurement to
gallons 10,080 x 44 gal
1 gal 231 in3
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Check It Out Example 3
The bath tub in Ravinas house measures 46 in.
by 36 in. by 24 in. Another bath tub with a
similar shape is smaller by a scale factor of
. There are 231 in3 in 1 gallon. Estimate how
many more gallons the larger bath tub holds.
1 2
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Check It Out Example 3 Continued
Rewrite the question as a statement.
Compare the capacities of two similar tubs, and
estimate how much more water the larger tub holds.
List the important information
The larger tub is 46 in. x 36 in. x 24 in.
231 in3 1 gal
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Check It Out Example 3 Continued
You can write an equation that relates the volume
of the small tub to the volume of the large tub.
The convert cubic inches to gallons to compare
the capacities of the tubs.
Volume of small tub
Volume of large tub (a scale factor)3
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Check It Out Example 3 Continued
Volume of large tub 46 x 36 x 24 39,744 in3
Volume of small tub 39,744 x 0.53 4,968 in3
Convert each volume into gallons
Subtract the capacities 172 gal 22 gal 150
gal
The large tub holds about 150 gallons more than
the small tub.
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Check It Out Example 3 Continued
Look Back
Halve the dimensions of the large tub and find
the volume 23 x 18 x 12 4,968 in3. Subtract
the volumes of the two tubs 39,744 4,968
34,776 in3. Convert this measurement to
gallons 34,776 x 150 gal.
1 gal 231 in3
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Lesson Quiz Part I

• Given the scale factor, find the surface area to
  the nearest tenths of the similar prism.
• 1. The scale factor of the larger of two similar
  triangular prisms is 8. The surface area of the
  smaller prism is 18 ft2.
• 2. The scale factor of the smaller of two similar
  triangular prisms is . The surface area of
  the larger prism is 600 ft2.

1,152 ft2
1 3
66.7 ft2
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Lesson Quiz Part II

• Given the scale factor, find the volume of the
  similar prism.
• 3. The scale factor of the larger of two
  similarrectangular prisms is 3. The volume of
  thesmaller prism is 12 cm3.
• 4. A food storage container measures 6 in. by 10
  in. by 2 in. A similar container is reduced by
  ascale factor of . Estimate how many
  moregallons the larger container holds.

324 cm3
1 2
about 0.5 gal
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Lesson Quiz for Student Response Systems
1. A fish storage aquariumcontainer measures 15
in. by 17 in. by 7 in. A similar container
aquarium is larger by a scale factor of 4.
Estimate how many more gallons the larger
container aquarium holds. A. about 494 gal
B. about 490 gal C. about 487 gal D.
about 486 gal
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Lesson Quiz for Student Response Systems
2. The volume of a prism is 28 cm3. What is the
volume of a similar prism that is larger by a
scale factor 4? A. 32 cm3 B. 112 cm3
C. 448 cm3 D. 1,792 cm3