Section 9.4 Volume and Surface Area

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 9.4

Volume and Surface Area


VolumeVolume is a measure of the capacity of a three-dimensional figure.It is the amount of space inside a three-dimensional figure.

Surface area is the sum of the areas of the surfaces of a three-dimensional figure.

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VolumeSolid geometry is the study of three dimensional solid figures.Volumes of three-dimensional figures are measured in cubic units such as cubic feet or cubic meters.Surface areas of three -dimensional figures are measured in square units such as square feet or square meters.

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Determine the volume and surface area of the following three dimensional figure.

Solution

Example 1: Volume and Surface Area

V lwh 35 2 3 30 ft

2 2 2SA lw wh lh 2 5 2 2 2 3 2 5 3 220 12 30 62 ft

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2 ft

3 ft

5 ft

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Determine the volume and surface area of the following three dimensional figure. When appropriate,use the π key on yourcalculator and roundyour answer to thenearest hundredths.

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Solution V r2h 42 8

SA 2 rh 2 r2

2 4 8 2 42

64 32 96 301.59 m2

402.12 m3 128

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Determine the volume and surface area of the following three dimensional figure. When appropriate,use the π key on yourcalculator and roundyour answer to thenearest hundredths.

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Solution

V

13 r2h

13 32 8

SA r2 r r2 h2

32 3 32 82

9 3 73 108.80 m2

75.40 m3 24

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Determine the volume and surface area of the following three-dimensional figure. When appropriate,use the π key on yourcalculator and roundyour answer to thenearest hundredths.

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Solution

V

43 r3

43 93

SA 4 r2

4 92

324 1017.88 cm2

3053.63 cm3 972

4 81

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PolyhedraA polyhedron is a closed surface formed by the union of polygonal regions.

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PrismA prism is a special type of polyhedron whose bases are congruent polygons and whose sides are parallelograms.The parallelogram regions are called the lateral faces of the prism.If all the lateral faces are rectangles, the prism is said to be a right prism.

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PrismThe prisms illustrated are all right prisms.When we use the word prism in this book, we are referring to a right prism.

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Volume of a Prism

V = Bh,

where B is the area of the base and h is the height.

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Example 5: Volume of a Hexagonal Prism Fish TankFrank’s fish tank is in the shape of a hexagonal prism. Use the dimensions shown in the figure and the fact that 1 gal = 231 in3 to answer the following:a) Determine the volume of the fish tank in cubic inches.

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Example 5: Volume of a Hexagonal Prism Fish Tank

Solution:The base is made up of two

congruent trapezoids, so:

2trap 1 2

1 1 (8)(16 8) 96 in2 2A h b b

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So, the area of the base is 2(96), or 192 in², and thus the volume is as follows:

3192(24)4608 . in

V Bh


Example 5: Volume of a Hexagonal Prism Fish Tankb) determine the volume of the

fish tank in gallons (round your answer to the nearest gallon).

Solution:

3

34608 in 1 gal

1 231 in4608 gal 231

V

19.95 gal9.4-18


PyramidA pyramid is a polyhedron with one base, all of whose faces intersect at a common vertex.

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Volume of a Pyramid

where B is the area of the base and h is the height.

V

13 Bh

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Example 6: Volume of a PyramidDetermine the volume of the pyramid.

SolutionArea of base = s2 = 22

= 4 m2

The volume is 4 m3.

V

13 Bh

13 4 3

4 m3

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Section 9.4 Volume and Surface Area