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Uchburchak yuzi

a  h_a
b  h_b
c  h_c 1



a²Sinβ Sinγ

1. 
   S    , S  abSinγ  ;
   

2 2 2 2 2Sinα

2. 
   S  p( p  a)( p  b)( p  c) (Geron formulasi);
   

3. 
   S  ^abc  pr ,
   
4. 
   S 
   

_{p } a  b  c <sub>;


, m </sub> m_a  m_b  m_{c .}
2

Uchburchakdagi asosiy teoremalar

1. 
   Sinuslar teoremasi:
   

a Sinα
_ b Sinβ
_ c Sinγ
 2R ,

R – tashqi chizilgan aylana radiusi.

2. Kosinuslar teoremasi:

a²  b²  c²  2bcCosα .

3. 
   a  bCosγ  cCosβ .
   

4. Molveyde formulasi: 5. Tangenslar teoremasi

a  b <sub>



Cos</sub>α  β
2 _.



a  b <sub>



tg</sub> α  β
2 _



ctg ^γ
2 _.

^c Sin ^γ
_{a  b tg} α  β _tg α  β

2
6. Sin^α  ;
2
2 2
Cos^α  .
2

Teng yonli uchburchak

h_c 
; S  ^c ;
2 4

_a2
R  ;
2h_{c
r } ^c ^{2a  c} _.
4h_c

1) а²  в²  с²

To‘g‘ri burchakli uchburchak

(Pifagor teoremasi);

2) Sinα  ^b , Cosα  ^a ; tgα  ^b , ctgα  ^a ;
c c a b
3) a²  ca^, b²  cb^;

h²  a^b^,
h  ^ab ;
с _c

4) R  ^c  m ,

_{r } a  b  c _,

r  R  ^{a  b} ;

2 ^c 2 2
ab ch a²tgα c²Sin2α ₂

5) S      r
2 2 2 4

• 
  2rR;
  

 ^d ²

c
6) _{ }
 
 ^b ,
a

 ^y ²

x
 
 
 ^b ,
a

l – bissektrisa.

R  ^3a , r  ^3a ,

Teng tomonli uchburchak

R  2r ;

3 6
h  3r  1,5 R  r  R,
S  ³ a² .
4

To‘rtburchaklar Kvadrat

P  4a ,
S  a^{2 d} ,



2
2

d  a ,
r  ^a ,
2
R  ^d .
2

To‘g‘ri to‘rtburchak

P  2(a  b) , S  ab ,

d  ,
R  ^d .
2

Parallelogramm

1 2
1) P  2(a  b) ; 2) 2a²  2b²  d ²  d ² ;
3) S  a  h  a  b  Sinα , S  ¹ d  d  Sinϕ .

a ₂ 1 2

1) 4a²  d ²  d


² ; 2)

Romb

S  a  h  2a  r  ¹ d  d ,

1 2


S  a²  Sinα ; 3)


h  2r .
₂ 1 2

Trapetsiya

1. 
   S  mh  ¹ d  d Sinϕ ;
   

₂ 1 2

2. 
   d₁  ab 
   

3) S₂  S₄ 
c²a  d ²b a  b ^;
,

S  _ 
_2 _.

4) AE=EC, BF=FD;

_{MN } a  b
2
EF  ^{a  b} ,
–o‘rta chiziq;
ME  FN  ^b ; ^AO  ^OD  ^a .

2 2 OC OB b

Teng yonli trapetsiya

^{b  a}²

x  ,
2
c²  h²  ;
4

S  ¹ d ²Sinα  ^{a  b} h ; 2r  h  .
2 2

To‘g‘ri burchakli trapetsiyada

1 2
d ²  d ²  a²  b² .

Aylanaga tashqi va ichki chizilgan to‘rtburchaklar

1. 
   Aylanaga tashqi chizilgan to‘rtburchak
   

a  c  b  d ;
S  pr  (a  c)r  (b  d )r , bu yerda 2 p  a 

2. Aylanaga ichki chizilgan to‘rtburchak

α  β  γ  φ  180;
ad  bc  d₁  d₂ ;
^{S }  ^{p  a}  ^{p  b} ^{p  c}  ^{p  d}  ^;

R  ¹ 4S
^{ab  cd} ^{ac  bd} ^{ad  bc} ^.

Ko‘pburchaklar

1. 
   ^{Ichki burchaklari yig‘indisi:} ^{n  2}^{π ;}
   
2. 
   Tashqi burchaklari yig‘indisi: 2π ;
   

3. 
   Diagonallari soni:
   

ⁿ^{n  3}
2

ta.

1. 
   Ichki burchagi: ^{n  2π}
   

n

2. 
   Tashqi burchagi: ^2π ;
   

n

Muntazam ko‘pburchaklar

;

3. 
   r 
   

;
2
r  n  a n  a

4) S_n  ₂  ₄ ;
5) a  2R  Sin^π  2r  tg ^π .
n n

Muntazam beshburchak, oltiburchak, sakkizburchak Muntazam beshburchak

Ichki burchaklari yig‘indisi: 540 ;
Ichki burchagi: 108 ; Tashqi burchagi: 72 ;

a  ^R
2
10  2 5  2r ;

R  ^a
 r _ 1_; r  ^R _  1_  ^a ;

10 4 10

_{d } 1 
a , d – diagonal;
S  ⁵ R²
_a2
 25  10 5  5r ² .

2 8 4

Muntazam oltiburchak

Ichki burchaklari yig‘indisi 720⁰;

Ichki burchagi: 120 ; Tashqi burchagi: 60 ;

a  R  ² r
3
; r  ^{a 3} ;
2
d₁  a ,
d₂  2R  2a ;

S  ³ R²  ³ a²
 2r ² .

2 2

Muntazam sakkizburchak

Ichki burchaklari yig‘indisi 1080 ;

Ichki burchagi: 135 ; Tashqi burchagi: 45;

a  R
R  ^a
2
 2r _
 r
1_;
;


r  ^R
2


_ a _

 1_;

d₁  a 2  2 ; d₂  a _1 
2 _;
d₃  2R  a ;

S  2R²  2a² _
 1_  8r² _
1_.