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MATHS PROJECT

WORK

SURFACE AREA AND VOLUME



Prepared By :- Mr. N. K . Saini



Submitted To : - K. V . SIKAR



Planning Ahead :-. !ube i. "ateral #ur$a%e area ii. Total #ur$a%e area

iii. Volume i&. '(ample# ). !uboid

i. "ateral Sur$a%e Area ii. Total #ur$a%e area iii. Volume



Planning Ahead :-*. !ylinder

i. !ur&ed #ur$a%e area ii. Total #ur$a%e areaiii. Volume

+. !one i. !ur&ed #ur$a%e area ii. Total #ur$a%e area

iii. Volume



Planning Ahead :-

,. Sphere i. Sur$a%e area

ii. Volume

. emi#phere

i. !ur&ed #ur$a%e area

ii. Total #ur$a%e area iii. Volume



Planning Ahead :-

/. 0ru#tum i. !ur&ed #ur$a%e area

ii. Total #ur$a%e areaiii. Volume

1.Sur$a%e area o$%ombination o$ #olid#

2. !on&er#ion o$ #olid#



!3B' :-

A %ube i# a threedimen#ional 4gure5

6ith #i( #ide#- allSide# in #hape o$

S7uare."ength o$ #ide i# denoted by

the letter 8l9.

l



"ateral Sur$a%e Area :-"ateral #ur$a%e area re$er# to the

area o$ only the 6all# it doe#not in%lude the area o$ the ;oorand roo$<.

0ormula :- + l= >eri&ation :- Sin%e all the #ide# o$

%ube are in the #hape o$ #7uare. area o$ the #7uare? l= no. o$ #ide# ?+

area ? +l=



'@AMP"'S :-.0ind the lateral #ur$a%e area o$

the %ube 6ith #ide o$ ,%m.

Sol.- e are gi&en-

l ? ,%m lateral #ur$a%e area ? +l=

? +, %m<=

? + )),%m= ? 2CC%m=



'@AMP"'S :-).0ind the lateral #ur$a%e area o$ the %ube

6ith area o$ one $a%e 1%m=. Al#o 4ndthe length o$ the #ide.

Sol. D Area o$ one $a%e ? 1%m=

l= ? 1%m=

l ? E1%m=

l ? 2%m

"ateral #ur$a%e area o$ %ube ? +l=

? +2%m<=

?+1%m=

? *)+%m=



'@AMP"'S :-*.0ind the #ide o$ the %ube 6ho#e lateral

#ur$a%e area i# 2%m= .Sol.- "ateral #ur$a%e area o$ %ube ?

2%m=

+l= ?2%m=

l= ?2%m=F+

l= ?+2%m=

l ?E+2%m=

l ? / %m



Total Sur$a%e Area G$

!ube :-Formula :- 6l²

Der!a"o# :- Sin%e all the $a%e# o$ a %ubeare #7uare# 5

Area o$ #7uare ? l=

No. o$ #7uare ?

Area o$ #7uare ? Total#ur$a%e area o$ %ube

? l= There$ore 5 total #ur$a%e area o$ the %ube i#

l= .



'@AMP"' :-. 0ind the total #ur$a%e area o$ the %ube

6ith #ide o$ /.)%m.Sol. - e are gi&en5

l ? /.)%m

Total #ur$a%e area ? l= ? /.)%m<=

? ,.1+ %m=

? *.C+ %m=



'@AMP"' :-).A gi$t in a #hape o$ %ube i# to be 6rapped in a

gi$t paper. 0ind the total %o#t o$ the 6rapperneed to %o&er the gi$t 6ho#e #ide i# .1%m5 atthe %o#t o$ R#., per m=.

Sol. D e are gi&en 5

#ide o$ the %ube l< ? .1 %m Total #ur$a%e area ? l<=

? .1 %m<=

? +.)+ %m=

? )//.++ %m= %o#t o$ the 6rapper ? R#. ,Fm=

? ,).//++m=

? R#. *.1/



Volume G$ !ube : -Volume o$ the %ube re$er# to the

#pa%e in#ide the #i( 6all#.

0ormula :- l l l

? lH 3nit :- unitH



'@AMP"' :-. Three e7ual %ube# are pla%ed

Side by #ide in a ro6. 0ind the&olume o$ the ne6 4gure $ormed5Al#o 4nd it# ratio in re#pe%t to the #ingle %ube.Sol.- "et 8a9 be the edge o$ ea%h %ube.

Volume o$ the #ingle %ube ? aH Sum o$ the &olume o$ three %ube# ? *aH

? *aHRatio o$ the &olume o$ t6o 4gure# ? Volume o$ the %ube F

Volume o$ the ne6 4gure ? aH F *aH ? :*

a

a

a

a



!3BGI> :-!uboid i# a three dimen#ional 4gure5

6ith #i( #ide# and all #ide# o$ e7ual length.In !uboid oppo#ite re%tangle# aree7ual.

It9# three dimen#ion# are :- ."engthl< ). Breadth b< *. eight h<

lb

h



"AT'RA" S3R0A!'

AR'A:-

"ateral #ur$a%e area o$ the %uboid re$er to the area o$

the $our 6all# o$ it.

0ormula :- )lb< h

>eri&ation :- Area o$ re%tangle ? lh

Area o$ re%tangle) ? bh Area o$ re%tangle* ? lh

Area o$ re%tangle + ? bh

Total area ?)lh)bh

? )lb< h

lb

h



TGTA" S3R0A!'

AR'A:-

0ormula :- )lb bh hl <

>eri&ation :- Area o$ re%tangle ? lh< Area o$ re%tangle ) ?lb < Area o$ re%tangle * ?lh < Area o$ re%tangle + ?lb < Area o$ re%tangle , ?bh <

Area o$ re%tangle ? bh <

? )lb < ) bh < ) lh < ? ) lb bh hl <

h



'@AMP"' :-1. Marry wants to decorate her Christmas tree. She wants to place her tree

on a wooden box covered with coloured paper with picture of Santa

clause on it . She must know the exact quantity of paper to buy it. If thedimensions of the box are : !cm" #!cm" $!cm% how many square sheetsof paper of side #!cm would she require&

Sol. '(he surface area of the box ) $*lb + bh + hl ,

) $- * !"#!, +*#!"$!, +*$!"!, ) $ */$!! + !! + 10!! , ) $ " 0!! cm2 ) 11$!! cm2

(he area of each sheet of paper) #!" #! cm3 ) 10!!cm3(herefore no. of sheets require ) Surface area of the box4 5rea of one sheet

of paper

) 11$!!4 10!! ) 6

(herefore % she would require 6 sheets.



'@AMP"' :-. A 6all o$ length Cm 6a# to be built a%ro## an open ground.the height o$ 6all i# +m and thi%Jne## o$ the 6all i# )+%m. I$ thi# 6all i# to

be built up 6ith bri%J# 6ho#e dimen#ion# are )+%m )%m 1%m5 ho6many bri%J# 6ould be re7uired Sol. D e are gi&en5 "ength ? Cm ? CCC%mBreadth ? )+%meight ? +m ? +CC%m

So5 &olume o$ 6all ? length breadth height? CCC )+ +CC%mHNo65 ea%h bri%J i# a %uboid 6ith length?)+%m5 breadth?)%m5 height?1%m

Volume o$ ea%h bri%J ? lbh ? )+ ) 1 %mHSo5 no. o$ bri%J re7uire ? &olume o$ the 6allF Volume o$ ea%h bri%J ? CCC )+ +CCF )+ ) 1 ? +.

So5 the 6all re7uire# +/ bri%J#.



!L"IN>'R :-

A right %ir%ular %ylinderi# a #olid generatedby the re&olution o$a re%tangle about

one o$ it# #ide.It i# a $olded re%tangle

6ith both %ir%ular

end#.

h

r



CURVED SURFACE AREA OF

CYLINDER:-!ur&ed #ur$a%e area o$ the %ylinder :-

? Area o$ the re%tangular#heet

? length breadth ? perimeter o$ the ba#e o$

the %ylinder h

? )r h ? )rh



EXAMPLE :-

. Shubhi had to maJe a model o$ a %ylindri%al Jaleido#%ope

$or her proe%t. She 6anted to u#e %hart paper to u#e %hartpaper to maJe the %ur&ed #ur$a%e o$ it. hat 6ould be thearea o$ %hart paper re7uired by her5 i$ #he 6anted to maJea Jaleido#%ope o$ length-),%m 6ith a *.,%m radiu#

Sol. D Radiu# o$ the ba#e o$ the %ylindri%al Jaleido#%ope r< ?

*.,%m

eight length< o$ Jaleido#%ope h< ?),%m

Area o$ paper re7uired ? %ur&ed #ur$a%e area o$

Jaleido#%ope ? )rh

? )))F/*.,), %m=

? ,,C %m=



TOTAL SURFACE AREA OF

CYLINDER :-Total #ur$a%e area o$ a %ylinder :

? area o$ the re%tangular #heet ) area o$ the%ir%ular region# <

? perimeter o$ the ba#e o$ %ylinder h ) area

o$ %ir%ular ba#e <? )rh )r=

? ) r r h <

$

r



EXAMPLE :-

. A barrel i# to be painted $rom in#ide and out#ide. It ha# no lid.The radiu# o$ it# ba#e and height i# .,m and )m re#pe%ti&e.0ind the e(penditure o$ painting at the rate o$ R#. 1 per #7uaremeter.

Sol. D Oi&en5 r? .,m 5 h ? )m Ba#e area o$ barrel ? r=Ba#e area to be painted in#ide and out#ide < ? ) r=

?) *.+ ., <= %m= ? ) *.+ ).), ?+.*%m=

!ur&ed #ur$a%e area o$ barrel ? ) rh Area to be painted ? ) ) rh ? + *.+ ., ) %m=

? ) *.+%m= ? */.1 %m=

Total area to be painted ? */.1 +.* < %m= ?,.1 %m=

'(penditure on painting ? R#. 1 ,.1? R#. ++.+1



VOLUME OF CYLINDER :-

Volume o$ a %ylinder %an be built up u#ing %ir%le#

G$ #ame #ie.So5 the &olume o$ %ylinder %an be obtained a# :- ba#e area height? area o$ %ir%ular ba#e height

? r=h

r



EXAMPLE :-

. A mea#uring ar o$ one liter $or mea#uring milJ i# o$

right %ir%ular %ylinder #hape. I$ the radiu# o$ the ba#ei# ,%m 5 4nd the height o$ the ar.Sol. D Radiu# o$ the %ylindri%al ar ? ,%m "et 8h9 be it# height Volume ? r=h

Volume ? liter ? CCC%mH Qr=h ? CCC ? CCCFr= ? CCC / F )),,

%m ? CCC/ F ))), %m ? +C F %m ?

)./* %m eight o$ the ar i# )./* %m .



EXAMPLE :-

. 0ind the 6eight o$ a hollo6 %ylindri%al lead pipe )%m long andF)%m thi%J. It# e(ternal diameter i# ,%m.eight o$ %mH o$ leadi# .+ gm <

Sol. D Thi%Jne## ? F)%m '(ternal radiu# o$ %ylinder ? R? )F)<%m ? ,F)%m Internal radiu# o$ %ylinder ? r ? ,F) D F) < ? ) %m Volume o$ lead ? R= - r= <h ? ,F)<= - )= )

? ))F/ ),F+ D + ) ? ))F/),-F+< ) ?2*F/ ? )1/F/ %mH

eight o$ %mH o$ lead ? .+ gm eight o$ %ylinder ? .+ )1/F/ gm ?

+/.1F/ gm ? )C2,.2/+ gm

? )C2,.2/+FCCC Jg ?).C2,2/+ Jg

? ).C2Jg

There$ore5 6eight o$ the %ylindri%al pipe i#).C2Jg



RIGHT CIRCULAR CONE :-

I$ a right angled triangle i# re&ol&ed about one

o$ it# #ide# %ontaining a right angle5 the #olid

Thu# $ormed i# %alled a right %ir%ular %one.

The point V i# the &erte( o$ %one.

The length GV?h5 height o$ the %one

The ba#e o$ a %one i# a %ir%le 6ith G a# %enter

and GA a# radiu#.

The length VA ? l 5 i# the #lant height o$ the%one.

V

l

h

G

r

A




CONE :-It i# the area o$ the %ur&ed part o$

the %one. '(%luding the %ir%ular ba#e <

0ormula :- F) perimeter o$ the ba#e #lant height

? )r l

? r l

l

r



XAMPLE :-

).o6 many meter# o$ %loth ,m 6ide 6ill be re7uired to

maJe a %oni%al tent 5 the radiu# o$ 6ho#e ba#e i#/m and 6ho#e height i# )+m

Sol. D Radiu# o$ ba#e ? /m

Verti%al height 5 8h9 ? )+m

Slant height 8l9 ? E h= r= ? E)+<= /<=

?E,/ +2 ? E), ? ), m

!ur&ed #ur$a%e area ? rl

? ))F/ /), m= ? ,,C m=

idth o$ %loth ? ,m

"ength re7uired to maJe %oni%al tent ? ,,CF, m

? Cm




CONE :-Total #ur$a%e area o$ the %one :-

?!ur&ed #ur$a%e area o$ %one %ir%ular ba#e

Red %oloured area green %oloured area <

?rl r=?r l r <

hh

hl

r



EXAMPLE :-

. Total #ur$a%e area o$ a %one i# //C%m=. I$ the #lant

height o$ %one i# + time# the radiu# o$ it# ba#e 5 then4nd the diameter# o$ the ba#e.

Sol. D Total #ur$a%e area o$ %one ? //C %m=

? r r l < ? //C

? l ? + radiu#? ? +r

? r r +r < ? //C

? ,r = ? //C

? r= ? //C / F , ))? / /

? r ? /%m

There$ore5 diameter o$ the ba#e o$ the %one

i# +%m.



VOLUME OF THE CONE :-

0ormula :- F* r=h

>eri&ation :- I$ a %ylinder and %one o$ #ane ba#e

Radiu# and height are taJen 5 and i$ %one i# put

3nder the %ylinder then it 6ill o%%upy only

Gne Dthird part o$ it .

There$ore5 &olume o$ %one i# F* o$ the &olume o$

!ylinder.

? F*r=h

hh

)

*

$

l

rr



EXAMPLE :-%& T$e ra'u( a#' )er)e#'*ular $e+$" o, a *o#e are # "$e ra"o :%.& , "$e

!olume o, "$e *o#e ( /%0*m12 3#' "( )er)e#'*ular $e+$" a#' (la#" $e+$"&

Sol& 4 Le" "$e ra'u( o, "$e *o#e 5 Per)e#'*ular $e+$" o, "$e *o#e 5 %.

Volume o, "$e *o#e 5 /%0 m1

He#*e2

%7/8r²$ 5 /%0

5 9r²$ 5 0.

5 /&%0 ;<² ;%.< 5 0.

5 / = /%0 1 5 0.

5 1 5 %

5 5 %

T$ere,ore2 )er)e#'*ular $e+$" o, "$e *o#e 5%.m

A#' ra'u( o, "$e *o#e 5m

Sla#" $e+$" o, *o#e 5 > r² ? $ ² 5 >² ?%.²

5 > . ? %00 5 >%6

5 %/m



EXAMPLE :-).A 6ooden right %ir%ular %one ha# a ba#e o$ radiu# *%m and height

+%m. The upper part o$ the %one %ut i# in #u%h a 6ay that the%oni%al pie%e 6ill ha&e height %m and ba#e radiu# C./,%m. 0indthe &olume o$ the remaining portion.

Sol. D 0or %omplete %one5 r ? *%m

height 8h9 ? +%m

Volume o$ the %omplete %one ? F*r=h

? F* * * + ? ) %mH0or the upper part o$ %one5 radiu# ? C./,%m 5 height ? %m

? F*R=

? F*C./,C./,

? C.1/, %mH

Volume o$ the remaining portion o$ the %one D

? Volume o$ the %omplete %oneD &olume o$ the %ut %one

? ) D C.1/, ? .1), ?.1), *.+

? */.C2 %mH



SPHERE : -

The #et o$ all point# in #pa%e e7uidi#tant $rom a 4(ed

point5 i# %alled a #phere .

The 4(ed point i# %alled the %enter o$ the #phere.

A line #egment pa##ing through the %enter o$ the #phere6ith it# end point# on the #phere i# %alled a diameter

o$ the #phere.

r





EXAMPLE :-

.I$ the diameter o$ a #phere i# 8d9 and %ur&ed

#ur$a%e area 8S95 then #ho6 that S ? d=.en%e5 4nd the #ur$a%e area o$ a #phere6ho#e diameter i# +.) %m.

Sol. D d ? )r

!ur&ed #ur$a%e area o$ #phere ? S

? +r= ? +r= ? )r<=

? d= ere5 d ? +.)%m

Sur$a%e area o$ the #phere ? d= ?))F/ +.)<=

? ,,.++%m=



VOLUME OF THE SPHERE :-

Volume o$ the #phere :-

?+F*RH



EXAMPLE :-

. o6 many #pheri%al bullet# %an be made out o$

lead 6ho#e edge mea#ure# ++%m5 ea%h bulletbeing +%m in diameter.

Sol. D "et the total no. o$ bullet# be (

Radiu# o$ #pheri%al bullet ? +F) %m ? )%m

Volume o$ a #pheri%al bullet ? +F* )<H %mH?+F* ))F/ 1 < %mH

Volume o$ #olid %ube ? ++<H %mH

Number o$ #pheri%al bullet# re%a#t ? &olume o$ %ube

? ++++++*/ &olume o$ one bullet

+ ))1

? **//

? ),+



EXAMPLE :-

). I$ the radiu# o$ a #phere i# doubled 5 6hat i# the ratio o$

the &olume o$ the 4r#t #phere to that o$ the #e%ond Sol. D 0or the 4r#t #phere 5

Radiu# ? r

Volume ? V

0or the #e%ond #phere5Radiu# ? )r

Volume ? V)

Then 5 V ? +F*rH

V) ? +F*)r<H ? +F*1rH< There$ore5 V ? +F*rH ?

V) ? +F*1H ? 1

Ratio ? :1

HEMISPHERE :



HEMISPHERE :-

A plane pa##ing through the %entre o$ a

#phere di&ide# the #phere into t6o e7ual

part# .

'a%h part i# Jno6n a# hemi- #phere.

r





HEMISPHERE :-

0ormula : - )r=

>eri&ation :-

Sin%e5hemi#phere i# hal$ o$ #phere-

There$ore5

Sur$a%e area o$ #phere ? +r=al$ o$ it ? )r=

r





HEMISPHERE :Total #ur$a%e area o$ hemi#phere:

? !ur&ed #ur$a%e area %ir%ular ba#e

? )r= r=

? *r=

EXAMPLE :-



EXAMPLE :-

. The internal and e(ternal diameter# o$ a hollo6 hemi#pheri%al &e##el are),%m and

)+ %m re#pe%ti&ely. The %o#t to paint %mH o$ the #ur$a%e i# R# C.C,. 0ind the total

%o#t to paint the &e##el all o&er .

Sol. D '(ternal area 6hi%h i# to be painted

? )R= ? )))F/),F)),F) %m=

? 1/,F/ %m=

Internal area 6hi%h i# to be painted ? )r= ? )))F/ )+F))+F) %m=

? **F/ %m=

Area o$ the ring at top ? ))F/ U),<= )+F) <=

? ))F/ ).,<= )< = ? ))F/ W)., )< ).,- )<? ))F/ )+., C.,

? )2.,F / %m= Total are to be painted? 1/, ** )2., ? *+1C., %m=

/ / / /

? 2),./1 %m=

!o#t o$ painting X Re. C.C,F%m= R#.

? R#. 2),./1 C.C, ? R#. 2.)12

? R# 2.)2

FRUSTUM : -



FRUSTUM : -

I$ a %one i# %ut by a plane parallel to the ba#e then the

part bet6een the ba#e and the plane i# %alled $ru#tum

o$ the %one

ere5 'BS0 i# $ru#tum $rom the %one AB!.

P0 ? R ? radiu#

Y! ? r ? radiu#

PY ? h ? height0! ? l ? #lant height

SURFACE AREA OF THE



SURFACE AREA OF THE

FRUSTUM :- Z!3RV'> S3R0A!' AR'A G0 0R3ST3M :-

? l R r < R= r=

l?E h= r D r <=

ZTGTA" S3R0A!' AR'A G0 0R3ST3M :-

? R r< l

l ? E h= R D r <=

R

rh

r

R

h l

EXAMPLE :-



EXAMPLE :-

.A $ri%tion %lut%h i# in the $orm o$ $ru#tum o$ a %one the

diameter being %m and C %m and length 1%m.0ind it# bearing #ur$a%e .

Sol. D "et ABB9A9 be the $ri%tion %lut%h

"et 8l9 be it# #lant height

l ? E 1= 1-,<= ? E + 2 ? E/* %m

Bearing #ur$a%e ? "ateral #ur$a%e area o$ ABB9A9

? l R r < ? ))F/ E/*1 , < %m=

? *+2 %m #7.

VOLUME OF THE FRUSTUM :-



VOLUME OF THE FRUSTUM :-

Volume o$ $ru#tum o$ %one :

? F* h R= r= Rr<

EXAMPLE :-



EXAMPLE :-

. The radii o$ the end# o$ a bu%Jet o$ height )+ %m are

, %m and , %m. 0ind it# %apa%ity.Sol. D !apa%ity o$ the bu%Jet ? Volume o$ the $ru#tum

? h r= R= Rr

*

? )) )+ ,<= ,= , , %mH

/ *

? )) 1 )), ), /,

%mH /

? / *), %mH

/

? 1/.+* %mH

VOLUMES OF COMBINATION



VOLUMES OF COMBINATION

OF SOLIDS :-

SGM' '@AMP"'S :-.The de%orati&e blo%J i# made up o$ t6o #olid# D a %ube

6ith edge ,%m and a hemi#phere 4(ed on the top

ha# a diameter o$ +.)%m. 0ind the total #ur$a%e area o$

the #olid.

Sol. D Total #ur$a%e area o$ %ube ? ,,

? ,C %m=

The #ur$a%e area o$ blo%J ? Total #ur$a%e area o$ blo%J %ur&ed#ur$a%e area o$ the hemi#phere D area o$ the ba#e o$ the

hemi#phere ? ,C - r= )r= ? ,C r=< %m=

? ,C %m= ))F/ +.)F) +.)F) %m=

? ,C *.1< %m= ? *.1 %m=

EXAMPLE



EXAMPLE :-). MayanJ made a bird Dbath $or hi# garden in the

#hape o$ a %ylinder 6ith a hemi#pheri%al depre##ion

at one end . The height o$ the %ylinder i# .+,mand it# radiu# i# *C%m . 0ind it# total #ur$a%e area.

Sol.- "et 8h9 be the height o$ the %ylinder

8 r9 be the %ommon radiu# o$ the %ylinder 5 and

hemi#phere Total #ur$a%e area o$ the bird bath ? !SA o$ %ylinder!SA o$ hemi#phere

? )rh )r= ?)r r h<

? )))F/ *C+,*C<%m=

? **CCC%m=

? *.*m =

EXAMPLE



EXAMPLE :-*.A #olid %on#i#ting o$ a right %ir%ular

%one o$ height )C%m and radiu# C%m

#tanding on a hemi#phere o$ radiu#C%m i# pla%ed in a right %ir%ular

%ylinder $ull o$ 6ater #u%h that it

tou%he# the bottom. 0ind the &olume o$ 6ater le$t in the

%ylinder i$ the radiu# o$ the %ylinder i# C%m and it# height i# 1C

%m.

Sol. D e are gi&en :-

eight o$ the %one 8h9 ? )C%m

Radiu# o$ the %one 8r9 ? C%m

Radiu# o$ the hemi#phere 8R9 ? C%m

Radiu# o$ the %ylinder 8R)9 ? C%m

eight o$ the %ylinder 8h9 ? 1C%m

EXAMPLE



EXAMPLE :-Volume o$ 6ater le$t ? Volume o$ %ylinder-&olume o$

%one &olume o$ hemi#phere<

? R)= h D F* r=h )F*RH < ? R= h - F* r=h )R)H<

? R=h D F* C C )C ) C C C <

? R=h D F*+*)CCC +*)CCC< ? R= h D F* 1+CCC ? R= -)11CCC<

? R=h D )11CCC< ? *CC 1C D)11CCC<

? +1CCC D )11CCC < ? *CCCC

? *.+ *CCCC ? *C+CC%mH

? *C+CC F CCCCCC

? .*mH

EXAMPLE :



EXAMPLE :-+. A #olid iron pole %on#i#t# o$ a

%ylinder o$ height ))C%m and ba#e

diameter )+%m5 6hi%h i# #urmounted by

another %ylinder o$ height C%m and

radiu# 1%m. 0ind the ma## o$ the pole5

gi&en that %mH o$ iron ha# appro(imately 1gm# ma##.

Sol.- >imen#ion o$ #maller %ylinder-

Radiu# 8 r9 ? 1%m

eight 8h9 ? C %m >imen#ion o$ large %ylinder-

Radiu# 8 R9 ? )%m

eight [9 ? ))C%m

Volume o$ the #tatue ? Volume o$ large %ylinder



Volume o$ the #tatue ? Volume o$ large %ylinder Volume o$

#mall %ylinder

? R= r=h

? *.+11C *.+))))C

? *.+ + C *.+ ++ ))C

? )CC2.C 2C.1++ ? )C,/. 22+/.) ?

,**.1%mH

Ma## o$ the #tatue-

Ma## o$ %mH ? 1gm#

Ma## o$ ,**.1 %mH ? ,**.1 1

? 12))C.+ gm#

In Jg- 12))C.+FCCC

? 12).) Jg#

FROM ONE FIGURE INTO



FROM ONE FIGURE INTO

OTHER :-

.A metalli% right %ir%ular %one )C%m high 6ho#eVerti%al angle i# C\ i# %ut into t6o part# at themiddle o$ it# height by a plane parallel to it# ba#e.I$ the $ru#tum #o obtained be dra6n into a 6ireo$ diameter F %m5 4nd the length o$ 6ire.

Sol.- ]AB! ?)C%m ^?C\ r? tan *C ?rF)C ? FE* ? rF)C ? r ? )CFE * %m

]A>' _ ? *C\ R ? tan *C ? RFC ? FE* ? RFC ? R ? CFE* %m 0ru#tum B!'> h ? C%m

Volume o$ the $ru#tum B!'> D



Volume o$ the $ru#tum B!'>

? F* h R= r= Rr <

? F* C )CFE*<= CFE* <= )CFE*<

CFE*< ?CF* )CCF* CCF* )CCF*< ? CF*/CCF* <

?C )) /CC

* / *

? ))CCCF/ %mH

Area o$ the 6ire :-

d ?F%m

r ? F*) %m

l ?

r= ? ))F/ F*) F*)

? ))F/1 %m=

"ength o$ the 6ire ? Volume o$ $ru#tum



"ength o$ the 6ire ? Volume o$ $ru#tum

Area o$ the 6ire

? ))CCCF2 %mH

))F/1 %m=

? ))CCC /1

2 ))

? /1CCCF2 %m

? /1C F2 m

? /2+.+ m

EXAMPLE :



EXAMPLE :-).o6 many #il&er %oin#5 ./,%m in diameter and

thi%Jne##)mm5

mu#t be melted to $orm a %uboid o$ dimen#ion,.,%mC%m*.,%m

Sol.-Radiu# o$ the %oin 8r9 ? ./,F) %m

eight o$ the %one 8h9 ? )mm ? C.)%m

Volume o$ %oin ? r=h ? ))F/ ./,F) ./,F)C.)

? C.+1%m

"ength o$ %uboid 8l9 ? ,.,%mBreadth o$ %uboid 8b9 ? C %m

eight o$ %uboid 8h 8 ? *.,%m

Volume o$ the %uboid ? l b h



Volume o$ the %uboid ? l b h

? ,., C *.,

? 2)., %mHNumber o$ %oin# needed ? &olume o$ %uboid

&olume o$ %oin

? 2)., CC

C.+1 CC

? +CC %oin#



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