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Volumes of Prismsand Cylinders

ObjectivesAfter studying this section, you will be able toa Find the volumes ofright rectangular prismsa Find the volumes of other prismsa Find the volumes of cylinders0 Use the area of a prism's or a cylinder's cross section to find the

solid's volume

Part One: IntroductionVolume of a Right Rectangular PrismThe measure of the space enclosed by a solid is called the solid'svolume. In a way, volume is to solids what area is to plane figures.

Definition The volume of a solid is the number of cubic unitsof space contained by the solid.

A cubic unit is the volume ofa cube with edges one unit long.A cube is a right rectangular prism with congruent edges, so all itsfaces are squares. InSection 12.1 we used the word box for a rightprism. Thus, a right rectangular prism can also be called a rectangular box.

One linear unit

U

One Cubic Unit

Each hy«r h^<- * ».-•.»<•

of 6 cubic units each

Z7 7

Zzz-r—7 27 T

2./

/

y

Rectangular Box

The rectangular box above contains 48 cubic units. The formula thatfollows is not only a way of counting cubic units rapidly but alsoworks with fractional dimensions.

Section 12.4 Volumes of Prisms and Cylinders 575

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Tt

th.

de

se<

Tl

C

w

a

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su

tv

se

D

se

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576

Postulate The volume of a right rectangular prism is equal tothe product of its length, its width, and its height

where € is the length, w is the width, and h is theheight.

Another way to think of the volume of arectangular prism is to imagine the prism tobe a stack of congruent rectangular sheets ofpaper. The area of each sheet is i • w, andthe height of the stack is h. Since the base of

•* .-I-.

there is a second formula for the volume ofa rectangular box.

y jf

& /w

1 v —A-'-

| = (€w)hj = (area ofsheet) •h

Theorem 115 The volume of a right rectangular prism is equal tothe product of the height and the area of the base.

Vrect.box=®hwhere 2& is the area of the base and h is the height.

Volumes of Other PrismsThe formula in Theorem 115 can be used to compute the volume ofany prism, since any prism can be viewed as a stack of sheets thesame shape and size as the base.

V = 9&h V = 2Bh V = 2&h

Theorem 116 The volume of any prism is equal to the product ofthe height and the area of the base.

Vprism = ®*where S& is the area of the base and h is the height.

Notice that the height of a right prism is equivalent to themeasure of a lateral edge.

Chapter 12 Surface Area and Volume

Problem 5 A box (rectangular prism) is sitting ina corner of a room as shown.

a Find the volume of the prism.b If the coordinates of point A in a

three-dimensional coordinate system are (4, 5, 6), what are the coordinates of B?

Solution a V = 0&h

= (4 • 5)6= 120

b Point C = (0, 0, 0) is the corner.To get from C to B, you wouldtravel 0 units in the x direction, 5units in the y direction, and 0units in the z direction. So thecoordinates of B are (0, 5, 0).

1 Find the volume of each solid.

a 12

2 Find the volume of cement needed to

form the concrete pedestal shown. (Leaveyour answer in it form.)

3 The area of the shaded face of the rightpentagonal prism is 51. Find the prism'svolume.

4 Find the volume and the total surfacearea of the rectangular box shown.

z-axis

f 5x-axis

10

(4, 5. 6)

B

y-axis

Section 12.4 Volumes of Prisms and Cylinders 579

Problem Set A, continued

5 a Find the volume of a cube with an edge of 7.

b Find the volume of a cube with an edge of e.

c Find the edge of a cube with a volume of 125. >-

6 Find the length of a lateral edge of a right prism with a volumeof 286 and a base area of 13.

Problem Set B

7 Traci's queen-size waterbed is 7 ft long,5 it wide, and 8 in. thick.

a Find the bed's volume to the nearestcubic foot.

b If 1 cu ft of water weighs 62.4 lb, whatis the weight of the water in Traci'sbed to the nearest pound?

8 If point A's coordinates in the three-dimensional coordinate system are (2, 5, 8),what are the coordinates of B?

9 Find the volume and the total area of theprism.

10 Find the volume and the total area of each right cylindrical solidshown.

12

11 When Hilda computed the volume and the surface area of acube, both answers had the same numerical value. Find thelength of one side of the cube.

Chapter 12 Surface Area and Volume

12 Find the volume and the surface area ofthe regular hexagonal right prism.

10

<z>

13 Find the volume ofa cube in which a face diagonal is 10.

14 A rectangular cake pan has a base 10 cm by 12 cm and a heightof8 cm. If810 cu cm ofbatter is poured into the pan, how far upthe side will the batter come?

15 A rectangular container is to be formedby folding the cardboard along the dotted lines. Find the volume of thiscontainer.

16 The cylindrical glass is full of water,which is poured into the rectangularpan. Will the pan overflow?

A17 Jim's lunch box is in the shape of a halfcylinder on a rectangular box. To thenearest whole unit, what is

a The total volume it contains?

b The total area of the sheet metaliioedeu to manufacture it?

18 A cistern is to be built of cement. Thewalls and bottom will be 1 ft thick. Theouter height will be 20 ft. The inner diameter will be 10 ft. To the nearest cubicfoot, how much cement will be neededfor the job?

i i

L6inN|

10 in

8 in

^Ji

Section 12.4 Volumes of Prisms and Cylinders 581

582

Problem Set C

19 A wedge of cheese is cut from a cylindrical block. Find the volume and the totalsurface area of this wedge.

20 An ice-cube manufacturer makes icecubes with holes in them. Each cubejp 4 cm on a side and the hole is 2 cmin diameter.

'd

a To the nearest tenth, what is the volume of ice in each cube?

b To the nearest tenth, what will be the volume of the water leftwhen ten cubes melt? (Water's volume decreases by 11% whenit changes from a solid to a liquid.)

c To the nearest tenth, what is the total surface area (includingthe inside of the hole) of a single cube?

d The manufacturer claims that these cubes cool a drink twiceas fast as regular cubes of the same size. Verify whether thisclaim is true by a comparison ofsurface areas. (Hint: The ratioof areas is equal to the ratio of cooling speeds.)

21 Find the volume of the solid at the right.(A representative cross section is shown.)

22 A cylinder is cut on a slant as shown.

Find the solid's volume.

Chapter 12 Surface Area and Volume

Top view

Volumes of Pyramidsand Cones

ObjectivesAfter studying this section, you will be able toa Find the volumes of pyramidsa Find the volumes of cones

b Solve problems involving cross sections of pyramids and cones

Part One: IntroductionVolume of a PyramidThe volume of a pyramid is related to thevolume of a prism having the same base andheight. At first glance, many people wouldguess that the volume of the pyramid is halfthat of the prism.

Such a guess would be wrong, though.The volume of a pyramid is actually one third ofthe volume of a prism with the same baseand height.

Theorem 119 The volume ofa pyramid is equal to one third ofthe productof the heightand the area of the base.

Vpyr=&hivkere <% is the aren of the base and h is Hie height.

The proofof this formula is complex and will not be shownhere.

Volume of a ConeBecause a cone is a close relative of a pyramid, although its base is circular rather thanpolygonal, the formula for its volume is similar to the formula for a pyramid's volume.

Section 12.5 Volumes of Pyramids and Cones 583

584

Theorem 120 The volume of a cone is equal to one third of theproduct of the height and the area of the base.

VCone =&h =l™2hwhere 28 is the area of the base, h is the height, andr is the radius of the base.

Cross Section of a Pyramid or a ConeUnlike a cross section of a prism or a cylinder, a cross section of apyramid or a cone is not congruent to the figure's base. Observe thatthe cross section parallel to the base is similar to the base in eachsolid shown below.

A

l-V

Cross Section of a Cone Cross Section of a Pyramid

The Similar-Figures Theorem (p. 544) suggests that in these solidsthe area of a cross section is related to the square of its distancefrom the vertex.

Theorem 121 In a pyramid or a cone, the ratio of the area of across section to the area of the base equals thesquare of the ratio of the figures' respectivedistances from the vertex.

$_ (k\2®= \h)

where $ is the area of the crosssection, 26 is thearea of the base, k is the distance from the vertexto the cross section, and h is the height of thepyramid or cone.

A proof of Theorem 121 is asked for in problem 15.

Part Two: Sample Problems

Problem 1

Solution

If the height of a pyramid is 21 andthe pyramid's base is an equilateraltriangle with sides measuring 8, whatis the pyramid's volume?

Vpyr = phSince the base is an equilateral triangle, 26 = ^V3 = I6V3.So V= \(l6V3)(21) = II2V3.

Chapter 12 Surface Area and Volume

Problem 2 Find the volume of a cone with abase radius of 6 and a slant heightof 10.

Solution Using the right triangle shown, wefind that the height of the cone is 8.

V = -irrzhvcone 3"1 "

=f*r(62)(8) =96tt

Problem 3

Solution

A pyramid has a base area of 24 sqcm and a height of 12 cm. A crosssection is cut 3 cm from the base.a Find the volume of the upper pyra

mid (the solid above the cross section).

b Find the volume of the frustum{the solid below the cross section).

a Since the cross section is 3 cm from the base, its distance, k, fromthe peak is 9 cm.

Vupper pyramid 3J" K

= 40.5 cu cm

To find the volume of the frustum, we subtract the volume of theupper pyramid from the volume ofthe whole pyramid.V, = V -V.,frustum ^ whole pyramid * upper pyramid

= |(24)(12) - 40.5= 96 - 40.5

= 55.5 cu cm

Part Three: Problem Sets

Problem Set A1 Find the volume of a pyramid whose

base is an equilateral triangle with sidesmeasuring 14 and whose height is 30.

2 Find, to two decimal places, the volumeof a cone with a slant height of 13 and abase radius of 5.

Section 12.5 Volumes of Pyramids and Cones 585

586

Problem Set A, continued

3 The pyramid shown has a square baseand a height of 12.

a Find its volume.

b Find its total area.

4 The volume of a pyramid is 42. If its base has an area of 14, whatis the pyramid's height?

5 Given: The right circular cone shown

Find: a Its volume

b Its lateral area

c Its total area

6 A tower has a total height of 24 m. Theheight of the wall is 20 m. The base is arectangle with an area of 25 sq m. Findthe total volume of the tower to the nearest cubic meter.

A well has a cylindrical wall 50 m deepand a diameter of 6 m. The tapered bottom forms a cone with a slant height of5 m. Find, to the nearest cubic foot, thevolume of water the well could hold.

Problem Set B

8 Find, to the nearest tenth, the volume ofa cone with a 60° vertex angle and aslant height of 12.

9 A pyramid has a square base with a diagonal of 10. Each lateral edge measures13. Find the volume of the pyramid.

Chapter 12 Surface Area and Volume

[

20m

6m

50

10 PABCD is a regular square pyramid.a Find the coordinates of C.

b Find the volume of the pyramid.

11 Find the volume remaining if the smallercone is removed from the larger.

12 A rocket has the dimensions shown. If60% of the space in the rocket is neededfor fuel, what is the volume, to the nearest whole unit, of the portion of therocket that is available for nonfuel items?

13 A gazebo (garden house) has a pentagonal base with an area of 60 sq m. Thetotal height to the peak is 16 m. Theheight of the pyramidal roof is 6 m. Findthe gazebo's total volume.

14 Use the diagram at the right to find

a x

b The radii of the circles

c The volume of the smaller cone

d The volume of the larger cone

e The volume of the frustum

z-axis

x-axis

50

16

15 Set up and complete a proof of Theorem 121. (Hint: First provethat the ratio of corresponding segments of a cross section and abase equals the ratio of h to k.)

Section 12.5 Volumes of Pyramids and Cones 587

588

Problem Set B, continued

16 Find the volume ofa cube whose total surface area is 150 sq in.

Problem Set C

17 A regular tetrahedron is shown. (Each ofthe four faces is an equilateral triangle.)Find the tetrahedron's total volume ifeach edge measures

a 6 b s

IS A regular octahedron (eight equilateralfaces) has an edge of 6. Find the octahedron's volume.

19 Find the volume of the pyramid shown.

20 Find the volume of the frustum shown.

Chapter 12 Surface Area and Volume

Volumes of Spheres

ObjectiveAfter studying this section, you will be able toa Find the volumes of spheres

Part One: Introduction

The following theorem can be proved with the help of Cavalieri'sprinciple (discussed in Problem Set D of this section), but we shallpresent it without proof.

Theorem 122 The volume of a sphere is equal to four thirds of theproduct of n and the cube of the radius.

"sphere ~~ ^where r is the radius of the sphere.

Some of the problems in this section require you to find boththe volumes and the surface areas of spheres. Recall from Section12.3 that the total surface area of a sphere is equal to 4ttt2.

Part Two: Sample Problem

Problem Find the volume of a hemisphere with a radius of 6.

Solution First we find the volume of a sphere with a radius of 6.

sphere s177

= |7T(6)d = 2887T

The hemisphere's volume is half that of the sphere. Thus,V hemisphere = 144tt, or «452.39.

Part Three: Problem Sets

Problem Set A

1 Find the volume of a sphere with

a A radius of 3 b A diameter of 18 c A radius of 5

Section 12.6 Volumes of Spheres 589

590

Problem Set A, continued

2 Find the volume and the surface area ofa sphere with a radius of 6.

3 Find the volume of the grain silo to thenearest cubic meter.

T15„

A plastic bowl is in the shape ofa cylinder with ahemisphere cut out. The dimensions are shown.a What is the volume of the cylinder?b What is the volume of the hemisphere?c What is the volume of plastic used to make the bowl?

What volume of gas, to the nearest cubic foot, is needed toinflate a spherical balloon to a diameter of 10 ft?

Problem Set B

6 A rubber ball is formed by a rubber shell filled with air. Theshell's outer diameter is 48 mm, and its inner diameter is 42 mm.Find, to the nearest cubic centimeter, the volume of rubber usedto make the ball.

7 Given: A cone and a hemisphere asmarked

Find: a The total volume of the solid

b The total surface area of thesolid

8 A hemispherical dome has a height of 30 m.a Find, to the nearest cubic meter, the

total volume enclosed.

b Find, to the nearest square meter, thearea of ground covered by the dome(the shaded area).

c How much more paint is needed topaint the dome than to paint the floor?

d Find, to the nearest meter, the radiusof a dome that covers double the areaof ground covered by this one.

Chapter 12 Surface Area and Volume

9 A cold capsule is 11 mm long and 3 mmin diameter. Find, to the nearest cubicmillimeter, the volume of medicine itcontains.

10 The radii of two spheres are in a ratio of 2:5.a Find the ratio of their volumes.

b Find the ratio of their surface areas.

11 A minisubmarine has the dimensions shown.a What is the sub's total volume?

b Knowing the sub's surface area is important in determining how much pressureit will withstand. What is the sub's totalsurface area?

Problem Set C12 An ice-cream cone is 9 cm deep and 4

cm across the top. A single scoop of icecream, 4 cm in diameter, is placed ontop. If the ice cream melts into the cone,will it overflow? (Assume that the icecream's volume does not change as itmelts.) Justify your answer.

13 The volume of a cube is 1000 cu m.

a To the nearest cubic meter, what is the volume of the largestsphere that can be inscribed inside the cube?

b To the nearest cubic meter, what is the volume of the smallestsphere that can be circumscribed about the cube?

14 Find the ratio of the volume ofa sphere to the volusmallest right cylinder that can contain it.

^tv

15 In the diagram, ABGH is a rectangle andAB = BC = CD. To the nearest wholenumber, what percentage of the area of00 is the area of ABGH?

J

(0, 10)Vaxis ABCD

\- ^>

x-axis

\(12.-2)HG FE(,5.-2,

Section 12.6 Volumes of Spheres 591

Problem Set C, continued

16 Compare the volumes of ahemisphere and a cone with congruent bases and equal heights.

Problem Set D

Plane m is a cross-sectional plane through solids A and B.

Cavalieri's Principle

If two solids Aand Bcan be placed with their bases coplanarand ifthe area of every cross section of Ais equal to the areaof the coplanar cross section of B, then Aand Bhave equalvolumes.

17 Show that the volume ofcylindrical shellAisequal to the volume ofcylinder Bbyusing Cavalieri's principle.

18 a Compare the cross-sectional areas of the solids shown below.b Use Cavalieri's principle to derive the formula for the volume

of a sphere. (Hint: Use the diagrams to find a formula for thevolume ofa hemisphere.)

Chapter12 SurfaceArea and Volume