section 9.4 volume and surface area

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

Section 9.4

Volume and Surface Area


What You Will Learn

VolumeSurface Area

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VolumeVolume is the measure of the capacity of a three-dimensional figure.It is the amount of material you can put inside a three-dimensional figure.Surface area is sum of the areas of the surfaces of a three-dimensional figure.It refers to the total area that is on the outside surface of the figure.

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VolumeSolid geometry is the study of three dimensional solid figures, also called space 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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r

Volume Formulas

Sphere

Cone

V = πr2hCylinder

V = s3Cube

V = lwhRectangular Solid

DiagramFormulaFigure

213V r h

343V r

l

h

w

s

s

s

h

h

r

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r

Surface Area Formulas

Sphere

Cone

SA = 2πrh + 2πr2Cylinder

SA= 6s2Cube

SA=2lw + 2wh +2lhRectangular Solid

DiagramFormulaFigure

SA r2 r r

2 h

2

SA 4r 2

l

h

w

s

s

s

h

r

h

r

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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 1136 198 ft3

SA 2lw 2wh 2lh 2 113 2 36 2 116 66 36 132 234 ft2

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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

1

3 r2h

1

3 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

4

3 r3

4

3 93

SA 4 r2

4 92

324 1017.88 cm2

3053.63 cm3 972

4 81

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Polyhedra

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

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Euler’s Polyhedron Formula

Number of

vertices

number of

edges

number of

faces= 2– +

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Platonic SolidA platonic solid, also known as a regular polyhedron, is a polyhedron whose faces are all regular polygons of the same size and shape.There are exactly five platonic solids.

Icosahedron:

20 faces,

12 vertices, 30 edges

Dodecahedron:

12 faces,


Octahedron:

8 faces,


Cube:

6 faces,


Tetrahedron:

4 faces,

4 vertices, 6 edges

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Prism

A prism is a special type of polyhedron whose bases are congruent polygons and whose sides are parallelograms.These 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 6: Volume of a Hexagonal Prism Fish TankFrank Nicolzaao’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 toa) determine the volume of the fish tank in cubic inches.

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Example 6: Volume of a Hexagonal Prism Fish TankSolutionArea of hexagonal base:two identical trapezoids

A

trap

1

2h b

1 b

2

A

trap

1

2(8)(16 8) 96 in2

Areabase = 2(96) = 192 in2

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Example 6: Volume of a Hexagonal Prism Fish TankSolutionVolume of fish tank:

V B h

192 24

4608 in3

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Example 6: Volume of a Hexagonal Prism Fish Tankb) determine the volume

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

Solution

V

4608

231 19.95 gal

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Pyramid

A 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

1

3Bh

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Example 8: Volume of a PyramidDetermine the volume of the pyramid.

SolutionArea of base = s2 = 22

= 4 m2

The volume is 4 m3.

V

1

3Bh

1

34 3

4 m3

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section 9.4 volume and surface area

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