Microsoft Word Formula kitobcha

Aylanadagi burchaklar

_{x } β  α
2
_{y } α  β
2

α  ^β
2
α  ^{β  γ} α 
2 2

^{a2 } ^{b  c} ^c

Aylanadagi teoremalar

PL  PK  PN  PM


ab  cd

AS  BS,
ASO  BSO

Doira, sektor, segment, halqa

1. 
   Aylana uzunligi:
   

Aylana va doira

l  2π R ;

2. 
   Doira yuzi:
   

S  π R²
π D²


;
4

3. 
   Yoy uzunligi:
   

^lёй
_ π Rα _.
180

Sektor yuzi Segment yuzi

π R²α π R²α 1 ₂

^Sсект ^
.
360
^Sсегм ^
360

• 
  R Sinα .
  

2

Aylananing ikki paralel vatarlari orasidagi bo‘lagi yuzi

^Sкес ^
π R²



360
^{β  α}  ^
1 _R2
2
^{Sinβ  Sinα}  ^.

Halqa yuzi

S_х  π _R²  r ² _ .

STEREOMETRIYA

Prizma

Ixtiyoriy prizma

Yon sirti:
S_ён  P_{п к}  l .

To‘la sirti:
S_Т  S_ён  2S_ас .

Hajmi: V
 S_п.к.  l  S_ac  h .

Diagonallar soni:
n(n  3) .

Bu yerda
S_п.к. – perpendikulyar kesim yuzi,
^Рп.к. ^-

perpendikulyar kesim perimetri.

To‘g‘ri burchakli parallelepiped:

Yon sirti: To‘la sirti: Hajmi: V
^{Sён  2}^{ac  bc}^{.
Sт  2}^{ab  ac  bc} ^.
 abc .

d ²  a²  b²  c² .
3 ta simmetriya tekisligiga ega.
8 ta uchi, 12 ta qirrasi, 6 ta yoqi, 4 ta dioganali bor.

Kub

Yon sirti:


S_ён  4а² .

To‘la sirti:
S_Т  6a² .

Hajmi:
V  a³ .

d  a
; r  ^a ; R  ^a
2 2
³ . 9 ta simmetriya tekisligiga ega.

Piramida Ixtiyoriy piramida

To‘la sirti:
S_T  S_ac  S_ён .

Hajmi:
V  ¹ S  h  ¹ S r .

₃ ас ₃ Т

Muntazam piramida

l –yon qirra, f – apofema.

P  n  a ,
_{S } nar _.

ас
S  ¹ P

ac

 f .

2
S  S Cosφ , φ –asosidagi ikki yoqli burchak.

ён ₂ ас
ас ён

 ^a ²
R²  _{ }
 ² 

 r ² ;

l²  R²  h² ;


f ²  r ²  h² .

Kesik piramida

S  S  S  S .
V  ¹ h_S   S _.

т 1 2 ён
₃ 1 2

Muntazam kesik piramida uchun:
^{S  1}  ^{P  P}  ^{ l , l -apofema.}

ён ₂ 1 2

Muntazam uchburchakli piramida

l – yon qirra, f – apofema,
α – ikki yoqli burchak.

H

f

.
 ^a2  ²
12
3
a² 3
2

a ₂
l   H .
3
1
a² 3

S_ён  ₂ af ,
S_ас  ₄ ,
V  ₃ S_ас  H  ₁₂ H .

Muntazam to‘rtburchakli piramida

l – yon qirra, f – apofema. f  .

l  .
r  ^a ,
6
R  ^a .
3

S_ён  2af
_ ^Sac _,
S_ас  a² ,
V  ¹

_a2
S_ас  H  H .

Cosφ
3 12

Silindr

R –asosining radiusi, H –balandlik.
S_ас  π R² .
S_ён  2π RH .
^{Sт.с.  2π R}  ^{R  H} ^.
V  π R²H .
Yon sirti yoyilmasi:

Konus

L -yasovchi, R -asosinig radiusi, H -balandlik.
L  .
S_ён  π RL .
^{Sт.с.  π R}  ^{R  L} ^.
V  ¹ π R²H  ¹ S d .
_{3 3} ён
Yon sirti yoyilmasining uchidagi

burchakni topish:
α  2π R / L .

Kesik konus

L -yasovchi,


R, r -asosining radiuslari, H -balandlik.

^{L }  ^{R  r} ^{2  H 2 . S  π L}  ^{R  r}  ^.
ён

^Sт.с.
^{ π} _^{R2  r 2  L} ^{R  r} _ ^.
V  ¹ π H _R²  Rr  r ² _.

3
Sfera va shar

R -radiusi, d -diametr.

S  4π R²  πd ² .
V  ⁴ π R³  ^π d ³ .
3 6

Shar segmenti

R -sharning radiusi, h -segment balandligi.

r  .
S_ён  2π Rh  π _r ²  h² _.

πh² π
^{Sт.с.  π} _^{2Rh  r 2} _^{. V } ^{3R  h} ^{ h}_^{3r2  h2} _^.
3 6

Shar sektori

^{Sт.с.  π R} ^{2h  r}  ^.

_{V } 2π
R²h  ^π d ²h .

3 6

Shar halqasi

S_ён  2π RH .
S  π _2RH  R ²  R ² _ . V  ¹ π H _3R ²  3R ²  H ² _ .

т.с. 1 2
₆ 1 2

Sharga ichki chizilgan konus

l -yasovchi, R -sharning radiusi,
H -konusning balandligi, r -radiusi.
r ²  H ²
R  . x  H  R .
2H

Konusga ichki chizilgan shar


_r RH _.
L  R

Tekislikda Dekart koordinatalar sistemasi To‘g‘ri chiziq tenglamasi

1. 
   To‘g‘ri chiziq tenglamasi y  kx  b , bu yerda k –
   

burchak koeffitsienti k  tgα , α – Ox o‘qi bilan hosil qilgan burchak;

2. 
   ^A1  ^{x1; y1}  ^{va A2}  ^{x2; y2}  ^{nuqtalardan o‘tuvchi to‘g‘ri}
   

chiziq tenglamasi:

y  y_{1 }

x  x_{1 ;}

y  k (x  x )  y ,

y  y x  x ^{1 1}
2 1 2 1

bunda k 
y₁  y_{2 ;}
x₁  x₂

3. 
   ^A ^{x1, y1}  ^{nuqtadan o‘tuvchi to‘g‘ri chiziq:}
   

y  y₁  k(x  x₁) ;

4. 
   Ikki to‘g‘ri chiziq orasidagi burchak tangensi:
   

tgφ 
k₁  k_{2 ;}
1  k₁  k₂

5. 
   Ikki to‘g‘ri chiziqning parallellik alomati:
   

k₁  k₂ ;

6. 
   Ikki to‘g‘ri chiziqning perpendikulyarlik alomati:
   

k₁  k₂  1;

7. 
   Ikki to‘g‘ri chiziqning kesishish alomati:
   

k₁  k₂ ;

8) A(x₁, y₁), B(x₂ , y₂ )
va С(x₃, y₃)
nuqtalarning bir to‘g‘ri chiziqda yotish sharti:

x₀  x_{1 }
x₂  x₀
y₀  y_{1 ;}
y₂  y₀

9) A(x₀ , y₀ )
nuqtadan
ax  by  c  0
to‘g‘ri chiziqqacha bo‘lgan masofa:

d  ;

10. 
    Parallel to‘g‘ri chiziqlar orasidagi masofa:
    

h  ;

10. 
    Tekislikda uchlari
    

A(x₁, y₁) ,
B(x₂ , y₂ )
va C(x₃, y₃)

nuqtalarda bo‘lgan ABC uchburchakning yuzi

S  (x
 x )( y  y )  (x  x )( y  y ) .

2 1 3 1 3 1 2 1

^Markazi ^a;b