Formule Geometrie [PDF]

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Y
   

c c c


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c  c   c c!"#$%cc

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c     c   c c    c   cc

cccc'cc'&cc %cc#&cc  c(    cc!"c c  ccc

(    !c cc cc"c c ''c c)c(  cc '!c!"#ccc )c  #&c c

Drep 



Drep 



re l

s la  e (n l

 e( n L

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Per e rl s a  rr la rlr ad a

P



 DDAsa P (Ll)

Ara drep ngl  lng ea la  ea

Adrep ng L AABCDAB

l& In cazl ns r BC&

Pa ra l Pa ra les e n drep ng care are

a ela rleegale salng eaegala c la  ea&

PABBCCDDAsaP* L  ndeLes ela rapa ra l (ABBCCDDAL)& Ara pa ra lla ra la ra  la ra2 2

adca Apa ra L & 2

In cazlns r AABCDAB &

rapezl Per e rl s a  rr la rlr adca

PAB  BC  CD  DA& Ara rapezl  (!aza mare  !aza mca) nal mea/2 adcaA rapez(B 

!)

/2  ar n cazlns r

AABCD(DC  AB)

AM/2  pen rca

DCB (!aza mare) AB! (!aza mca) ar AM (nal mea)&

Cercl AvemOA - raza (n & r) Lngmeacercl (crcmfern acercl) ±




 

Ara cercl (crec ar f ara dscl) 




   

c

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    c   c

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c

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c + cc-c c!"  &c #c ,c-c c  &c#c ,-c-c  c(   c &c#c ,)c-c( c cc( c (   c &c(#c -)c-c( c!" c &c!#&c  c cc c  cf  c   c ccccc !c c(#$%&c  c!" c ccccc!!c c!#$%c c!c  c (  c!" &c  c  cc c!" cc c cc , c cccccc ,( !c c#$.&c   ( cfc  cc (   c !"cc c  c+ - c&c

c c

Paralelppedldrep ngc Avem AB - lngme(n & L) BC - la me(n & l) AE - nal measamcala erala (n & ) Ara la erala  smaarlrfe elrla erale Ala P!



ndeP!es eperme rl!aze sa Ala 2(L  l)



Ara !aze A!L

l&

Ara  ala  ara !aze  ara la erala Vlml VparalelppedA! VparalelppedL

 l

&

Paralelppedldrep ngces e n caz par clar de prsma arc!les e n caz par clar de paralelppeddrep ngc n senslcaes e n paralelpped c

a ela rlecngren e& De aceea n amn mnmcdespreeleac&

rncl de pramda Avem AB - Mca !aze mar A'B' - Mca !aze mc OO' - Inal me AA' - Mca la erala OM - Ap ema !aze mar O'M' - Ap ema !aze mc MM' - Ap ema rncl de pramda (n & a ) Ara la erala  sma arlr fe elr la erale Ala (PBP!)a /2 nde P! es e perme rl !aze mc ar PB es e perme rl !aze mar& Arle !azelr se calcleaza n fnc e de na ra !azelr ar la pramda regla a se ma p calcla s c aj rl frmlelr A!P! a!& ABPB aB& Ara  ala  ara !aze mar  ara !aze mc  ara la erala Vlml

•

V rnc de pramda
       

c c


 
 

c c c

c

 c + cc 'c-c    c &c#c --'c-c   c &c/cc " cc c  c ( c + #cc -c-c "!" c &c #cc  c!" cc c c cc!"c cc c :


 c

 c cc ccccc:


   c

 c  0cc : 
 
  c , c cc  c c


   c

Cnl Avem VA - genera are (n & g) VO - nal meacnl (n & h) AO - raza!aze (n & r) Ara !aze  ara cercl de la !aza adca      Ara la erala      Ara  ala     
  Vlml cnl      

rnchl de cn Avem A'A - genera are (n & ) OO' - nal mea rnchl de cn (n & I) AO - raza!azemar(n & R) A'O' - raza!azemc(n & r) Ara la erala   
  Ara  ala        Vlml  
     

c

§fera Avem OA - razã (n & r) Ara sfere      Vlml sfere