heron's formular surface areas and volumes & triangles

7/21/2019 Heron's Formular Surface Areas and Volumes & Triangles

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EFTNBEFTOHX

NB\M@X CM\ERIF

O@T\MDRHTOM@Qb frb cfeoiofr wotn tnb snfpbs mc ef`y pf`b himsbd coaurbs suhn fs squfrbs, rbhtfàibs, qufdroiftbrfis, roant

trofàibs, bquoiftbrfi trofàibs, osmshbibs trofàibs, shfib`b trofàibs, bth. Qb k`mw tnb ruibs tm co`d tnb pbroebtbrs

f`d frbf mc smeb mc tnbsb coaurbs. Cmr bxfepib, f rbhtfàib wotn ibàtn 6< e f`d wotn > e nfs pbroebtbr bqufi tm

< (6 ) 5 4:e. Tnb frbf mc tnos rbhtfàib os bqufi tm (6< >) e< 5 ?3 e


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EFTNBEFTOHX

Cmr bxfepb, f roant fàibs trofàib nfvoà twm sodbs mc ibàtn 8 e f`d 1 e (mtnbr tnf` tnb nypmtbùsb), nfs ots

frbf 5: e f`d h 5 2 62 5 12 e

Ot os aovb` tnft pbroebtbr 5 42:


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EFTNBEFTOHX

Bx.4 Tnb ibàtns mc tnb sodbs mc f trofàib frb 2 he, 6< he f`d 68 he. Co`d tnb ibàtn mc pbrpb`dohuifr crme tnb

mppmsotb vbrtbx tm tnb sodb wnmsb ibàtn os 68 he.

Xmi. Nbrb, f 5 2, l 5 6< f d h 5 68.

s 5 he


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EFTNBEFTOHX

Bx.1 O` coa. FLHD os f cobid o` tnb cmre mc f qufdroiftbrfi wnmsb sodbs frb o`dohftbd o` tnb coaurb.

OcDFL 5 ?::, co`d tnb frbf mc tnb cobid.Xmi. Hibfriy,DFL os f roant-fàibd trofàib. Tnbrbcmrb,

DL< 5 DF< + FL< [Rsoà Zytnfamrfs TnbmrbeU

DL< 5 ?< + 4:6

5 46e

O`DFL, wb nfvb


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EFTNBEFTOHX

Xmi. Ibt FLHD lb tnb aovb` trfpbzoue o` wnohn FL 5


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EFTNBEFTOHX

5 :::>:6>: ee

Nb`hb, bfhn mc tnb twm hnoidrb` woii abt f` frbf mc 4>:: e : de nfs tnb dofam`fi bqufi tm 31< de. Co`d ots frbf tm tnb `bfrbst squfrb ebtrb.

66. F trfpbzoue wotn ots pfrfiibi sodbs o` tnb rftom 63 7 62 os hut crme f rbhtfàib wnmsb sodbs ebfsurb 38 e f`d 2 e

rbspbhtovbiy. Tnb frbf mc tnb trfpbzoue os62

4mc tnb frbf mc tnb rbhtfàib. Co`d tnb ibàtns mc tnb pfrfiibi sodbs mc

tnb trfpbzoue.

6


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EFTNBEFTOHX

MLGBHTOSB T_ZB VRBXTOM@X

6. Tnb lfsb mc f roant fàibs trofàib Os - 2 ebtrbs f`d nypmtbùsb os 68 ebtrbs. Ots frbf woii lb

(F) >.>: (L) \s. 182 (H) \s. 18:.>: (D) \s. 1


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(F) )%( 6::


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EFTNBEFTOHX

F`s. H L L F H D L L F L

Vub. 66 6< 68 64 62 63 61 6> 6?


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EFTNBEFTOHX

XR\CFHB F\BFX & SMIREX

O@T\MDRHTOM@X

Rp toii `mw wb nfvb lbb` dbfioà wotn coaurbs tnft hf` lb drfw` m` tnb pfab mc mur `mtblmmk mr m` tnb

lifhklmfrd. Tnbsb frb hfiibd p i f ` b c o a u r b s . O` tnos hnfptbr, wb snfii study flmut smeb smiod coaurbs iokb hulmods,

hulb, hyio`dbr f`d spnbrb. Tnbsb coaurbs frb t n r b b d o e b ` s o m ` f i c o a u r b s . Qb snfii fism ibfr` tm co`d tnb surcfhb

frbfs f`d vmiuebs mc tnbsb coaurbs.

R@OTX MC EBFXR\BEB@T MC F\BF F@D SMIREB

Tnb o`tbr-rbiftom`snops lbtwbb` vfromus uòts mc ebfsurbeb`t mc ibàtn, frb f`d - vmiueb frb iostbd lbimw cmr

rbfdy rbcbrb`hb 7

IB@ATN

6 Hb`toebtrb (he) 5 6: eoioeobtrb (ee)

6 Dbhoebtrb (de) 5 6: hb`toebtrb6 Ebtrb (e) 5 6: de 5 6:: he 5 6::: ee

6 Dbhfebtrb (dfe) 5 6: e 5 6::: he

6 Nbhtmebtrb (ne) 5 6: dfe 5 6:: e

6 Koimebtrb (ke) 5 6::: e 5 6:: dfe 5 6: ne

6 Eyrofebtrb 5 6: Koimebtrb

F\BF

6 he< 5 6 he 6 he 5 6: ee 6: ee 5 6:: ee> he> he>

< 1

>

r 564

r 5 6

< 5 he8 5 ?.2< he8

Nb`hb, tnb vmiueb mc tnb hm`b 5 ?.2< he 8.


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Bx.


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EFTNBEFTOHX

Cmreuifb

Cmr f spnbrohfi snbii wotn bxtbr`fi rfdous 5 \ f`d o`tbr`fi rfdous 5 r, wb nfvb 7

Tnohk`bss m snbii 5 (\ - r) u`ots.

Mutbr surcfhb frbf 5


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EFTNBEFTOHX

Smiueb mc tnb eftbrofi 5 8

Qnft ibàtn mc tfrpfuio` 8 e wodb woii lb rbquorbd tm efkb hm ohfi tb`t mc nboant > e f`d lfsb rfdous 3 e=

fssueb tnft tnb bxtrf ibàtn mc eftbrofi tnft woii lb rbquorbd cmr stothnoà efrao`s f`d wfstfab o` huttoà

os fpprmxoeftbiy


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EFTNBEFTOHX

. F pimt mc if`d o` tnb cmre mc f rbhtfàib nfs doeb`som` : e. F drfo` ibt 6: e wodb os dua fii frmu`d ot

(m` tnb mutsodb) f`d tnb bfrtn dua mut os bvbìy sprbfd mvbr tnb pimt, o`hrbfsoà ots surcfhb ibvbi ly he 83 he. Crme bfhn m`b mc ots hmr`brs, f

squfrb mc > he os hutmcc. F` mpb` lmx os efdb mc tnb rbefoòà snbbt. Co`d tnb vmiueb mc tnb lmx.

6:. Qftbr o` f hf fi, 8: de wodb f`d 6< de dbpp, os cimwo a wotn f vbimhoty mc a.

6


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EFTNBEFTOHX

6>. F wmmdb tmy os o` tnb cmre mc f hm`b suremu`tbd m` f nbeospnbrb. Tnb dofebtbr mc tnb lfsb mc tnb hm`b os 3 he

f`d ots nboant os 4 he. Co`d tnb hmst mc pfo`to`a tnb tmy ft tnb rftb mc \s. 2 pbr 6::: he


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EFTNBEFTOHX

6?. Tnb crm t hmepmu d wfii mc f nmusb os dbhmrftbd ly wmmdb` spnbrbs mc dofebtbr


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6


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EFTNBEFTOHX

MLGBHTOSB T_ZB VRBXTOM@X

6. F rbhtf`auifr sf`d lmx os 2 e wodb f`d < e im`a. Nmw ef y huloh ebtrbs mc sf`d frb `bbdbd tm coii tnb lmx uptm

dbptn mc 6: he =

(f) 6 (l) 6: (h) 6:: (d) 6:::


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(f) \s. 8?3 (l) \s. 4:: (h) \s. 46: (d) \s. 4


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6


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EFTNBEFTOHX

(F) 3 he (L) 1 he (H) > he (D) ? he


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EFTNBEFTOHX

XR\CFHB F\BFX & SMIREBX F@XQB\ KB_ B^B\HOXB

(F) SB\_ XNM\T F@XQB\ T_ZB VRBXTOM@X 7

6.< ( + l)n 8

26 e8


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EFTNBEFTOHX

Vub


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EFTNBEFTOHX

trofàibs frb bqufi o.b. sox bqufiotobs nmid ammd, tnrbb mc tnb hmrrbspm`doà sodbs f`d tnrbb mc tnb hmrrbspm`doà

fàibs.

O` tnos sbhtom`, wb snfii prmvb tnft oc tnrbb prmpbriy hnmmsbs hm`dotom`s mut mc tnb sox hm`dotom`s frb sftoscobd, tnb`

tnb mtnbr tnrbb frb futmeftohfiiy sftoscobd. Ibt us `mw doshuss tnmsb hm`dotom`s wnohn b`surb tnb hmàrub`hb mc

twm fàibs.


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EF@OXNKREF\

EFTNBEFTOHX

O. XODB-F@AIBX(XFX)HM@A\RB@HB H\OTB\OM@

FÔME 7 Twm trof`taibs frb hmàrub`t oc twm sodbs f`d o`hiudbd fàib mc m`b trofàib frb bqufi tm tnb

sodbs f`d tnb o`hiudbd fàib mc tnb mtnbr trofàib.

O` tnb aovb` coaurb, o` FLH f`d DBC, wb nfvb 7

FL 5 DB, FH 5 DC f`d .DF

DBCFLH [Ly XFX-hrotbrofU

\BEF\K 7 XFX-Hmàrub`hb ruib nmids lut FXX mr XXF mr XXF ruib dmbs `mt nmid.

OO. F@AIB (FXF) HM@A\RB@HB H\OTB\OM@

TNBM\BE-6 7 Twm trofàibs frb hmàrub`t, oc twm fàibs f`d tnb o`hiudbd sodb mc tnb m`b trofaib frb bqufitm twm fàibs f`d tnb o`hiudbd sodb mc tnb mtnbr trofàib.

Aovb` 7Twm trofàibs FLH f`d DBC suhn tnft CHBL , f`d LH 5 BC.

Tm prmvb 7 .DBCFLH

Zrmmc 7 Tm prmvb .DBCFLH wb `bbd tm hm`sodbr tnrbb pmssolib sotuftom`s 7

Hfsb 6. Ibt FL 5 DB.

XTFTBEB@T \BFXM@

6.


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EFTNBEFTOHX

XTFTBEB@T \BFXM@6.

:

Xoeoifriy. CBD :6>:

Aovb`Aovb`Crme(8)Ly FXF hrotbrof


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EFTNBEFTOHX

OS XODB-XODB-XODB (XXX) HM@A\RB@HB H\OTB\OM@

Twm t r o f ` a i b s f r b h m ` a r u b ` t , o c t n b t n r b b s o d b s m c m ` b t r o f ` a i b f r b b q u f i t m t n b h m r r b s p m ` d o ` a t n r b b s o d b s

m c tn b m tn b r tr o f ` a i b .

Oc tnb aovb` coaurb, o`FLH f`dDBC, wb nfvb 7

FL 5 DB, LH 5 BC f`d FH 5 DC.

FLHDBC [Ly XXX-hrotbrofU

S \OANT F@AIB-N_ZMTB@RXB-XODB (\NX) HM@A\RB@HB H\OTB\OM@

T w m r o a n t f ` a i b s f r b h m ` a r u b ` t , o c t n b n y p m t b ` u s b f ` d m ` b s o d b m c m ` b t r o f ` a i b f r b r b s p b h t o v b i y b q u f i t m

t n b n y p m t b ` u s b f ` d m ` b s o d b m c t n b m t n b r t r o f ` a i b .

O` tnb aovb` coaurb,FLH f`dDBC frb roant - f`aibdtrof`aibs o` wnohn Nyp. FH 5 Nyp. DC f`d LH 5 BC.

FLHDBC [Ly \NX-hrotbrofU

Bx.6 O` coa. MF 5 ML f`d MD 5 MH. Xnmw tnft [@HB\TU

(o) FMD LMH f`d (oo) FD || LH.Xmi. Aovb` 7 O` coa. MF 5 ML f`d MD 5 MH.

Tm prmvb 7(o)FMD LMH f`d (oo) FD || LH.

Zrmmc 7

Nb`hb, prmvbd.

Bx.< O@ FLH, FL 5 FH. Oc Z os f pmo`t mc FL f`d V osf pmo`t m` FH suhn tnft FZ 5 FV. Zrmvb tnft

(o) FZH FVL (oo) LZH HVL.Xmi. Aovb`7 O`FLH, FL 5 FH. Z os f pmo`t m` FL f`d V os

f pmo`t m` FH suhn tnft FZ 5 FV.

XTFTBEB@T \BFXM@

6.


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EFTNBEFTOHX

Tm prmvb 7 (o)FZH FVL (oo)LZH HVL.

Zrmmc 7

Nb`hb, prmvbd.

Bx.8 O` coaurb, FH 5 FB, FL 5 FD f`dLFD 5BFH, prmvb tnft LH 5 DB. [@HB\TUXmi. Aovb` 7 O` coaurb, FH 5 FB, FL 5 FD f`dLFD 5BFH.

Tm prmvb 7LB 5 DB.

Hm`struhtom` 7 Gmo`t DB.

XTFTBEB@ \BFXM@

6.


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Nb`hb, prmvbd.

Bx.4 O` coaurb, dofam`fi FH mc f qufdroiftbrfi FLHD losbhts tnb fàibs F f`d H. Zrmvb tnft FL 5 FD f`d HL 5

HD/

Xmi. Aovb` 7 O` coaurb, dofam`fi FH mc f qufdroiftbrfi FLHD

losbht tnb fàibs F f`d H.

Tm prmvb 7 FL 5 FD f`d HL 5 HD

Nb`hb, prmvbd.

Bx.2 FL os f io`b sbaeb`t f`d Z os ots eod-pmo t. D f`d B frb pmo ts m` tnb sfeb sodb mc FL suhn tnftLFD 5FLB f`dBZF 5DZL. Xnmw tnft (o) DFZ BLZ (oo) FD 5 LB. [@HB\TU

Xmi. Aovb` 7 FL os f io`b sbaeb`t f`d Z os ots eod-pmo`t. D f`d B frb pmo`ts mc tnb sfeb sodb mc FL suhn tnftLFD 5

FLB f`dBZF 5DZL.

XTFTBEB@ \BFXM@

6.

8.

8.

O`FLH f`dFDB

(o) FL 5 FD

(oo) FH 5 FB

(ooo)LFD 5BFH

LFD +DFH 5DFH +BFH

LFH DFB

FLH FDB

LH 5 DB

Aovb`

Aovb`

Aovb`

FddoàDFH tm lmtn sodbs

Ly XFX hrotbrof

H.Z.H.T.

XTFTBEB@ \BFXM@

6.


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Tm prmvb 7(o)DFZ BLZ(oo) FD 5 LB

Zrmmc 7

Nb`hb, prmvbd

Bx.3 FL os f io`b-sbaeb t. Fs f`d Ly frb twm bqufi io`b - sbaeb ts drfw` m` mppmsotb sodbs m` io`b FL suhn

tnft F^ || L_. oc FL f`d ^_ o`tbrsbht bfhn mtnbr ft Z. Zrmvb tnft 7

(o) FZ^ LZ_ (oo) FL f`d ^_ losbht bfhn mtnbr ft Z.

Xmi. Aovb` 7 FL os f io`b-sbaeb`t. F^ f`d L_ frb tm bqufi io`b-sbaeb`ts drfw` m` mppmsotb sodbs mc io`b FL suhn tnft

F^ || L_.

Tm prmvb 7(o)FZ^ LZ_ (oo) FL f`d ^_ losbht bfhn mtnbr ft Z.

XTFTBEB@ \BFXM@

6.


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Nb`hb, prmvbd.

Bx.1 O` coa., FD os f ebdof` f`d LI, HE frb pbrpb`dohuifrs drfw` crme L f`d H rbspbhtovbiy m` FD f`d FD

prmduhbd. Zrmvb tnft LI 5 HE.

XTFTBEB@ \BFXM@

6.


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EFTNBEFTOHX

Xmi. Aovb` 7 O` coa., FD os f ebdof` f`d LI, HE frb pbrpb`dohuifrs drfw` crme L f`d H rbspbhtovbiy m` FD f`d FD

prmduhbd.

Tm prmvb 7 LI 5 HE

Nb`hb, prmvbd.

Bx.> FD f`d LH frb bqufi pbrpb`dohuifrs tm f io`b sbaeb`t FL. Xnmw tnft HD losbhts FL. [ @HB\TU

Xmi. Aovb` 7 FD f`d LH frb bqufi pbrpb`dohuifrs tm f io`b sbaeb`t FL.

Tm Zrmvb 7HD losbhts FL.

Nb`hb, prmvbd.

Bx.? Oc coa.LHD 5 FDH f`dFHL 5LDF. Zrmvb tnft FD 5 LH f`dF 5L/

XTFTBEB@ \BFXM@

6.


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Xmi. Aovb` 7 O` coaLHD 5FDH f`dFHL 5LDF

Tm prmvb 7(o) FD 5 LH (oo)F 5L

Nb`hb, prmvbd.

Bx.6: O` tnb coa. FLHD os f qufdroiftbrfi o` wnohn FL 5 FD f`d LH 5 DH. Zrmvb tnft 7

(o) FH losbhts bfhn mc tnb f`aibs F f`d H.

(oo) LB 5 BD.

(ooo)FLH 5FDH. Hf` wb sfy tnft FB 5 BH =Xmi. Aovb` 7 O` tnb coa. FLHD os f qufdroiftbrfi o` wnohn

FL 5 FD f`d LH 5 DH.

Tm prmvb 7 (o) FH losbhts bfhn mc tnb f`aibs F f`d H.

(oo) LB 5 BD.

(ooo)FLH 5FDH. Hf` wb sfy tnft FB 5 BH =

XTFTBEB@T \BFXM@

6.

::

Hmeem`

Crme (6)

Aovb`

Aovb`

Ly FFX hrotbrof

H.Z.H.T.


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@m. FB BH. Crme FB tm lb bqufi tm BH, ot os `bhbssfry tnftFBD HBD cmr wnohn< eust lb bqufi tm4.

Nb`hb, prmvbd.

Bx.66 FL os f io`b sbaeb t, Z f`d V frb pmo ts m` mppmsotb sodbs mc FL suhn tnft bfhn mc tnbe os bquodostf t

crme tnb pmo`ts F f`d L. Xnmw tnft tnb io`b ZV os tnb pbrpb`dohuifr losbhtmr mc FL. [@HB\TU

Xmi. Aovb` 7 FL os io`b sbaeb`t. Z f`d V frb pmo`ts m` mppmsotb sodbs mc FL suhn tnft bfhn mc tnbe os bquodostf`t crme

tnb pmo`ts F f`d L.

Tm prmvb 7ZV os tnb pbrpb`dohuifr losbhtmr mc FL.

XTFTBEB@T \BFXM@

6.


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Crme (3) f`d (1) wb hm`hiudb tnft ZV os tnb pbrpb`dohuifr losbhtmr mc FL.

Nb`hb, prmvbd.

XTFTBEB@T \BFXM@

6.

::

::

FHZ 5 ?::

Aovb`

Aovb`

Hmeem`

Ly XXX hrotbrof

H.Z.H.T.

Aovb`

FZV 5LZV, crme (8)

Hmeem`

Ly XFX hrotbrof

H.Z.H.T.

Io`bfr pfor mc f`aibs


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Bx.6


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EFTNBEFTOHX

Nb`hb, prmvbd.

Bx.64 FD, LB f`d HC, tnb fitotudbs mc FLH frb bqufi. Zrmvb tnft FLH os f` bquoiftbrfi trofàib.

Xmi. Aovb` 7 FD, LB f`d HC frb tnb fitotudbs mcFLH f`d frb bqufi.

Tm prmvb 7FLH os f` bquoiftbrfi trofàib

Nb`hbFLH os f` bquoiftbrfi trofàib.

Bx.62 O` f FLH, tnb o`tbr`fi losbhtmrs mcL f`d H ebbt ft M. Zrmvb tnft MF os tnb o`tbr`fi losbhtmr mcF.

XTFTBEB@T \BFXM@

6.


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Xmi. Aovb` 7 O` fFLH, tnb o`tbr`fi losbhtmrs mcL f`dH ebbt ft M.

Tm prmvb 7MF losbhtsF.

Hm`struhtom` 7 Drfw MD LH, MB, HF f`d MC FL.


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EFTNBEFTOHX

Zrmmc 7

Nb`hb, prmvbd.

Z\MZB\TOBX MC F@ OXMXHBIBX T\FO@AIB

O` tnos sbhtom`, wb woii ibfr` smeb prmpbrtobs rbiftbd tm f trofàib wnmsb twm sodbs frb bqufi. Qb k`mw tnft f

trofàib wnmsb twm sodbs frb bqufi os hfiibd f` osmshbibs trofàib. Nbrb, wb woii fppiy XFX hmàrub`hb hrotbrof f`d

FXF (mr (FFX) hmàrub`hb hrotbrof tm study smeb prmpbrtobs mc f` osmshbibs trofàib.

TNBM\E - 8 7 Fàibs mppmsotb tm bqufi sodbs mc f` osmshbibs trofàib frb bqufi.

Aovb` 7FLH os f` osmshbibs trofàib f`d FL 5 FH.

Tm prmvb 7 L 5H.Hm`struhtom` 7 Drfw FD tnb losbhtmr mcF. FD ebbts LH ft D.Zrmmc 7

XTFTBEB@T \BFXM@

6.

.

O`MDH f`dMBH(o)MHD 5MHB

(oo)MDH 5MBH

(ooo) MH 5 MH

MDH MBH

MD 5 MB

MD 5 MC

MB 5 MC

O` roantMBF f`dMCF

(o) MB 5 MC

(oo) Nyp. MF 5 Nyp, MFMBF MCF

MFB 5MFC

MF losbhtsF.

MH losbhtsH.

Bfhn 5 ?:: (ly hm`struhtom`).

Hmeem`

Ly FFX hrotbrof

H.Z.H.T.

Xoeoifriy, (MDL MCL).

Crme (8) f`d (4)

Crme (2)

Hmeem`

Ly \NX hrotbrof

H.Z.H.T .


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EFTNBEFTOHX

Nb`hb, prmvbd.

HM\MIIF\_ bfhn fàib mc f` bquoiftbrfi trofàib os mc 3::.

Aovb` 7FLH os f` bquoiftbrfi trofàib.

Tm prmvb 7 F 5L 5H 5 3::.Zrmmc 7

Nb`hb, prmvbd.

TNBM\BE-4 7 Tn b so d b s m p p m so tb tm b q u f i f ` a i b s m c f tr o f ` a i b b q u f i .

Aovb` 7FLH o` wnohnL 5H.

XTFTBEB@T \BFXM@

6.

:: 5 3::

FLH os f` bquoiftbrfi trofàib.

FL 5 FH

FL 5 LH

Crme (


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EFTNBEFTOHX

Tm prmvb 7FL 5 FH.

Hm`struhtom` 7 Drfw FD LH. FD ebbts LH o` D.

Zrmmc 7

Nb`hb, prmvbd.

Bx.63 O` tnb fdgmoòà coa, co d tnb vfiub mc x.

Xmi. Qb nfvb,

HFD +FDH +DHF 5 6>:: [Fàib sue prmpbrtyU

HFD +FDH + 34: 5 6>::

HFD +FDH 5 (6>:: - 34:) 5 663:

Lut HD 5 HFHFD 5FDH [s mppmsotb tm bqufi sodbs mc ffrb bqufiU

Xm, HFD 5FDH 5:

@mw,FDH 5FLD +DFL [Bxt.mc f5 sue mc o`t. mpp.sU

Lut, FD 5 LDFLD 5DFL.

Xm, FDH 5


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Bx.61 FLH os f osmshbibs trofàib os wnohn FL 5 FH. Xodb LF os prmduhbd tm D suhn tnft FD 5 FL. Xnmw tnftLHD os f roant fàib.

Xmi. Aovb` 7FLH os f` osmshbibs trofàib os wnohn FL 5 FH. Xodb LF os prmduhbd tm D suhn tnft FD 5 FL

Tm prmvb 7LHD 5 ?::

Zrmmc 7

Nb`hb, prmvbd.

Bx.6> O` f roant fàibd trofàib, m`b fhutb fàib os dmulib tnb mtnbr. Zrmvb tnft tnb nypmtbùsb os dmulib tnb

sefiibst sodb.

Xmi. Aovb` 7 FFLH o` wnohnL 5 ?:: f`dFHL 5 ::

::

LHD 5 ?::

LHD os f roant fàib.

Aovb`

s mppmsotb tm bqufi sodbs mc ffrb bqufi

Aovb`Crme (6) & (


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Zrmmc 7

Nb`hb, prmvbd.

O@BVRFIOTOBX O@ F T\OF@AIB

TNBM\BE-2 7 O c tw m so d b s m c f tr o f ` a i b f rb u ` b q u f i , tn b ` tn b a rb f tb r so d b n f s a rb f tb r f ` a i b m p p m so tb tm o t .

Aovb` 7FFLH o` wnohn FH 0 FL.

Tm prmvb 7FLH 0FHL.

Hm`struhtom` 7 Efrk f pmo`t D m` FH suhn tnft FD 5 FL. Gmo` LD.

Zrmmc 7

Nb`hb, prmvbd.

XTFTBEB@T \BFXM@

6.


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TNBM\BE-3 (Hm`vbrsb mc Tnbmrbe-2) 7 Oc twm fàibs mc f trofàibs frb u`bqufi, tnb` tnb arbftbr fàib nfs

arbftbr sodb mppmsotb tm ot.

Aovb` 7FFLH o` wnohnFLH 0FHL.Tm prmvb 7 FH 0FL.

Zrmmc 7

Nb`hb. prmvbd.

TNBM\BE-1 7 Tnb sue mc f`y twm sodbs mc f trofàib os arbftbr tnf` ots tnord sodb.

Aovb` 7FFLH.

Tm prmvb 7 (o) FL + FH 0 LH

(oo) FL + LH 0 FH

(ooo) LH + FH 0 FL.

Hm`struhtom` 7 Zrmduhb LF tm D suhn tnft FD 5 FH. Gmo` HD.

Zrmmc 7

XTFTBEB@T \BFXM@6. Qb efy nfvb tnrbb pmssoloiotobs mìy

(o) FH 5 L(oo) FH 9 FL(ooo) FH 0FL

Mut mc tnbsb bxfhtiy m`b eust lb trub.Hfsb-O. FH 5 FL.

FLH 5FHL.Tnos os hm`trfry tm wnft os aovb`.

FH 5 FL os `mt trubHfsb -OO FH 9 FL

FL 0 FHFHL 0FLHTnos os hm`trfry tm wnft os aovb`FH 9 FL os `mt trub.Tnus, wb frb ibct wotn tnb mìy pmssoloioty

FH 0 FL wnohn eust lb trub.

s mppmsotb tm bqufi sodbs mc ffrb bqufi

Arbftbr sodb nfs arbftbr fàib mpp. tm ot.


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Nb`hb, prmvbd.

\BEF\K 7 (o) Tnb ifrabst sodb mc f trofàib nfs tnb arbftbst fàib mppmsotb tm ot f`d hm`vbrsb os fism trub(oo) Tnb sefiibst sodb mc trofàib nfs tnb sefiibst fàib mppmsotb tm ot f`d hm`vbrsb o` fism trub

Bx.6? O` coa, snmw tnft 7 (o) FL 0 FH (oo) FL 0 LH f`d (ooo) LH 0 FH.

Xmi. Aovb` 7 FFLH o` wnohnL 5 4:: f`dFHD 5 6:::.

Tm prmvb 7 (o) FL 0 FH

(oo) FL 0 LH

(ooo) LH 0 FH.

Zrmmc 7

Nb`hb, prmvbd.

Bx.:: f`dF 5 3::

Arbftbr fàib nfs arbftbr sodb mpp. tm ot.

F 5 3:: f`dL 5 4::

Arbftbr fàib nfs arbftbr sodb mpp. tm ot.


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Zrmmc 7

ZE os tnb snmrtbst mc tnb io b sbaeb`ts crme Z tm FL.

Nb`hb, prmvbd.

Bx.


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TNO@AX TM \BEBLB\

6. Twm coaurbs frb hmàrub`t , oc tnby frb mc tnb sfeb snfpb f`d mc tnb sfeb sozb.. Fàib mppmsotb tm bqufi sodbs mc f trofàib frb bqufi.

?. Xodbs mppmsotb tm bqufi fàibs mc f trofàib frb bqufi.

6:. Bfhn fàib mc f` bquoiftbrfi trofàib os mc 3::.

66. Oc tnrbb sodbs mc m`b trofàib frb bqufi tm tnrbb sodbs mc tnb mtnbr trofàib, tnb` tnb twm trofàibs frb

hmàrub`t (XXX hmàrub`hb ruib).

6


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HLXB LFXBD XMEB OEZM\TF@T VRBXTOM@X

V.6 O` coa, DZ 5 LV,DZL 5 LVD f`dFDZ 5HLV, Xnmw tnft FDZ HLV.

[No`t 7DZL 5LVD

6>:: -DZL 5 6>:: -LVDFZD 5HVLU

V.< f`d e frb twm pfrfiibi io`bs o`tbrsbhtbd ly f`mtnbr pfor mc pfrfiibi io`bs p f`d q. Xnmw tnft

FLH HDF. [@HB\TU

[No t7 (o)LFH 5DHF

(oo)FHL 5HFD

(ooo) FH 5 HFUV.8 \fe wosnbs tm dbtbreo`b tnb dostf`hb lbtwbb` twm mlgbhts F f`d L, lut tnbrb os f` mlstfhib lbtwbb`

tnbsb twm mlgbhts fs snmw` o` coa, wnohn prbvb`ts nos crme efkoà f dorbht ebfsurbeb`t. Nb dbvosbs f`

oàbòmus wfy tm mvbrhmeb tnos doccohuitiy. Corst, nb coxbs f pmib ft f hm`vbòb`t pmo`t M sm tnft crme M,

lmtn F f`d L frb vosolib. Tnb` nb coxbs f`mtnbr pmib ft tnb pmo`t D m` tnb io`b FM (prmduhbd) suhn tnft FM

5 DM. O` f soeoifr wfy, nb coxbs f tnord pmib ft tnb pmo`t H m` tnb io`b LM (prmduhbd) suhn tnft LM 5 HM.

Tnb` nb ebfsurbs HD f`d co`ds tnft HD 5 61: e. Zrmvb tnft tnb dostf`hb lbtwbb` tnb mlgbht F f`d L os

fism 61: e.

V.4. O` roant trofàib FLH, roant fàibd ft H, E os tnb eod-pmo t mc nypmtbùsb FL. H os gmo bd tm E f`d

prmduhbd tm f pmo`t D suhn tnft DE 5 HE. Zmo`t D os gmo`bd tm pmo`t L. Xnmw tnft 7

(o) FEH LED(oo) DLH os f roant fàib.(ooo) DLH FHL


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(ov) HE 5::

DLH 5 ?::

(ooo) Ly XFX hmàrub`hb ruib,DLH FHL

(ov)DLH FHLHD 5 FL

FEH LED HE 5 DEHE 5

. O` tnb fdgmoòà coaurb, prmvb tnft 7

(o) FL + LH + HD 0 DF

(oo) FL + LH + HD + DF 0


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?. O` tnb fdgmo`o a coaurb, M os tnb hb trb mc f horhib, ^_ os f dofebtbr f`d ^W os f hnmrd. Zrmvb tnft ^_ 0 ^W.

6:. O` tnb aovb` coaurb, FD 5 FL f`d FB losbhtsF. Zrmvb tnft 7

(o) LB 5 BD

(oo) FLD 0LHF.

(H) @HB\T VRBXTOM@X 7

6. O` qufdroiftbrfi FHLD, FH 5 FD f`d FL losbhtsF. Xnmw tnftFLH FLD. Qnft hf` ymu sfy flmut LH f`d

LD =


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(o) FZL FVL

(oo) LZ 5 LV mr L os bquodostf t crme tnb fres mcF.

4. O`FLH, FD os tnb pbrpb`dohuifr losbhtmr mc LH. Xnmw tnftFLH os f` osmshbibs trofàib o` wnohn FL 5 FH.

2. FLH os f` osmshbibs trofàib o` wnohn fitotudbs LB f`d HC frb drfw` tm bqufi sodbs FH f d FL rbspbhtovbiy. Xnmw

tnft tnbsb fitotudbs frb bqufi.

3. FLH os f roant fàibd trofàib o` wnohnF 5 ?:: f`d FL 5 FH. Co`dL f`dH.

1. FD os f` fitotudb mc f` osmshbibs trofàib FLH o` wnohn FL 5 FH. Xnmw tnft 7

(o) FD losbhts LH

(o) FD losbhtsF.

>. LB f`d HC frb twm bqufi fitotudbs mc f trofàib FLH. Rsoà \NX hmàrub`hb ruib, prmvb tnft tnb trofàib FLH os

osmshbibs.

?. FLH os f` osmshbibs trofàib wotn FL 5 FH. Drfw FZ LH tm snmw tnftL 5H.

6:. FL f`d HD frb rbspbhtovbiy tnb sefiibst f`d imàbst sodbs mc f qufdroiftbrfi FLHD. Xnmw tnftF 0H f`dL

0D.

66. D os f pmo t m` sodb LH f`dFLH suhn tnft FD 5 FH. Xnmw tnft FL 0 FD.


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(D) QNOHN MC TNB CMIIMQO@A XTFTBEB@T FB T\RB (T) F@D QNOHN F\B CFIXB (C) 7

(f) Fàibs mppmsotb tnb bqufi sodbs mc f trofàib frb bqufi.

(l) Bfhn fàib mc f` bquoiftbrfi trofàib os 3::

(h) Xodb mppmsotb tnb bqufi fàibs mc f trofàib efy lb u`bqufi.

(d) Tnb losbhtmrs mc tnb twm bqufi fàibs mc f trofàib frb bqufi.

(b) Twm roant trofàibs frb hmàrub t, oc nypmtbùsb f`d f sodb mc m`b trofàib frb rbspbhtovbiy bqufi tm tnb

nypmtbùsb f`d f sodb mc tnb mtnbr trofàib.

(c) Oc f y twm sodbs mc roant trofàib frb rbspbhtovbiy bqufi tm twm sodbs mc mtnbr roant trofàib, tnb` tnb twm

trofàibs frb hmàrub`t.

(a) Tnb twm fitotudbs hmrrbspm`doà tm twm bqufi sodbs mc f trofàib `bbd `mt lb bqufi.

(n) Oc twm sodbs f`d f` fàib mc f trofàib frb rbspbhtovbiy bqufi tm twm sodbs f`d f` fàib mc mtnbr trofàib,

tnb twm trofàibs frb hmàrub`t.

(o) Oc twm sodbs f`d tnb o hiudbd fàib mc f trofàib frb rbspbhtovbiy bqufi tm twm sodbs f d tnb o hiudbd fàib

mc tnb mtnbr trofàib, tnb twm trofàibs frb hmàrub`t.

(g) Oc twm fàibs f`d tnb o hiudbd sodb mc m`b trofàib frb rbspbhtovbiy bqufi tm tnb twm fàibs f`d tnb

o`hiudbd sodb mc tnb mtnbr trofàib, tnb twm trofàibs frb hmàrub`t.

(k) Oc FLH Z\V, tnb` FL 5 ZV.(i) Oc twm fàibs f`d f sodb mc f trofàibs frb rbspbhtovbiy bqufi tm twm fàibs f`d f sodb mc tnb mtnbr trofàib,

tnb` tnb twm trofàibs frb hmàrub`t.

(e) Oc DBC \ZV, tnb`D 5V.

(`) Oc ZV\ HFL, tnb` ZV 5 HF.(m) Oc twm sodbs mc f trofàib frb u`bqufi, tnb` ifrabr sodb nfs tnb sefiibr fàib mppmsotb tm ot.

(p) Oc twm fàibs f trofàib frb u`bqufi, tnb` tnb arftbr fàib nfs tnb ifrabr sodb mppmsotb tm ot.

(q) Xue mc f`y twm sodbs mc f trofàib os arbftbr tnf` tnb tnord sodb.

(r) Doccbrb`hb mc f`y twm sodbs mc f trofàib os bqufi tm tnb tnord sodb.

(s) Mc fii tnb io`b sbaeb ts tnft hf` lb drfw` crme f pmo t tm f io`b `mt hm`tfoò a ot, tnb pbrpb`dohuifr io b

sbaeb`t os tnb snmrtbst.(t) Xue mc tnb tnrbb sodbs mc f trofàib os ibss tnf` tnb sue mc ots tnrbb fitotudbs.

(B) COII O@ TNB LIF@KX 7

(f) Xodbs mppmsotb tnb bqufi fàibs mc f trofàibs frb ]]]]].

(l) Fàibs Mppmsotb tnb bqufi sodbs mc f trofàibs frb ]]].

(h) O` f` osmshbibs trofàib FLH wotn FL 5 FH, oc LD f`d HB frb ots fitotudbs, tnb` LD os ]]]] HB.

(d) Oc fitotudbs HB f`d LC mc f trofàib FLH frb bqufi, tnb` FL 5 ]]]].

(b) O` roant trofàibs ZV\ f`d DBC, oc nypmtbùsb ZV 5 nypmtbùsb BC f`d sodb Z\ 5 DB, tnb`

ZV\ ]]]].

(c) O` f trof aib FLH, oc LH 5 FL f`dH 5 >::, tnb`L 5 ]]]].

(a) O` f trofàib ZV\, oc Z 5\, tnb` ZV 5 ]]]].

(n) Oc twm sodbs f`d tnb ]]] fàib mc m`b trof aib frb rbspbhtovbiy bqufi tm twm sodbs f`d tnb o hiudbd fàib mctnb mtnbr trofàib, tnb` tnb trofàibs frb hmàrub`t.

(o) Oc ]]]] sodbs mc f trofàib frb rbspbhtovbiy bqufi tm tnb tnrbb sodbs mc tnb mtnbr trofàib, tnb` tnb trofàibs

frb hmàrub`t.

(g) Oc o` twm trofàibs FLH f`d ZV\, FL 5 V\, F V f`dL 5\, tnb`FLH ]]]].

(k) Oc o` twm trofàibs FLH f`d DBC, FL 5 DC, LH 5 DB f`dL 5D, tnb`FLH ]]]].

(i) Oc o` twm trof aibs ZV\ f`d DBC, Z\ 5 BC, V\ 5 DB f`d ZV 5 CD, tnb`ZV\ ]]]]].(e) Xue mc f`y twm sodbs mc f trofàib os ]]]] tnf` tnb tnord sodb.(`) Oc twm fàibs mc f trofàib frb u bqufi. tnb tnb sefiibr fàib nfs tnb ]]] sodb mppmsotb tm ot.(m) Mc fii tnb io`b sbaeb ts drfw` crme f pmo t tm f io`b `mt hm`tfoòà ot, tnb ]]] io`b sbaeb`t o` tnb snmrtbst.(p) Doccbrb`hb mc f`y twm sodbs mc f trofàib os ]]]] tnf` tnb tnord sodb.


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(q) Oc f`y twm sodbs mc f trofàib frb u`bqufi, tnb` tnb ifrabr sodb nfs tnb ]]] fàib mppmsotb tm ot.(r) Tnb sue mc tnb tnrbb fitotudbs mc f trofàib os ]]]] tnf` ots pbroebtbr.(s) O` f roant trofàib, tnb nypmtb usb os tnb ]]] sodb.

(t) Tnb pbroebtbr mc f trofàib os ]]] tnf` tnb sue mc ots ebdof`s.MLGBHTOSB T_ZB VRBXTOM@X

6. Cmr f trofàib FLH, tnb trub stftbeb`t os 7

(F) FH< 5 FL< + LH< (L) FH 5 FL + LH (H) FH 0 FL + LH (D) FH 9 FL + LH

::

(L) 83::

(H) ?::

(D)


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(D) FL FZ

6:. Tnb ebfsurbs mc tnrbb fàibs mc f trofàib frb o` tnb rftom 6 7 < 7 8 Tnb` tnb trofàib os 7

(F) \oant fàibd (L) Bquoiftbrfi (H) Osmshbibs (D) Mltusb fàibd

66. Tnb nypmtbùsb mc f roant fàibd trofàib os


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6>. M`b sodb mtnbr tnf` tnb nypmtbùsb mc f roant fàibd osmshbibs trofàib os 4 he. Tnb ibàtn mc tnb pbrpb`dohuifr m`

tnb nypmtbùsb crme tnb mppmsotb vbrtbx os 7

(F) > he (L)


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Oepmrtf`t @mtbs

heron's formular surface areas and volumes & triangles

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