G e o metri y a planimetriya
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geometriya malumotnoma
Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 102 · Asosiga ichki chizilgan aylana radiusi r ga asosidagi ikki yoqli burchagi j ga teng bo'lgan muntazam piramidaga shar r radiusli shar ichki chizilgan bo'lsa: 1
sin r r cos j j = × + . Muntazam uchburchakli piramida l – yon qirrasi, f – apofema, a - ikki yoqli burchak. · Yon sirti: 3 2 yon S a f = × ; · To’la sirti: ( )
2 3 12 4 t a S a a h = + + ; · Hajmi: 2 1 3 3 12
a V S H H = = , 2 3 4 asos a S = ; · 2 2 2 2 12 a f H r H = + = + , 2 2 2 2 3
l H R H = + = + ; · 3 6 a r × = , 3 3 a R × = . Muntazam tetraedr a - tetraedrning har bir qirrasi. · Yon sirti: 2 3 3 4 yon S a = ; · To’la sirti: 2 3 t S a = × ; · Hajmi: 3 2
a V = ; · 2 3 a H = , 3 4
H = , 6 4
R = , 3 R r = ; · 4
r = , 6 12
r = . Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 103 Muntazam to'rtburchakli piramida l - yon qirrasi uzunligi, h - apofemasi, H - balandligi, a - asosining tomoni uzunligi, φ - ikki yoqli burchak. · Yon sirti: 2
S a h = , asos yon S S cos j = ; · To’la sirti: t asos yon S S S = + , 2
S a = ;
· Hajmi: 2 1 3 V a H = , 2 2 2 a H l = - ; · 2 2 2 2 4 a h H r H = + = + , 2 2 2 2 2
l H R H = + = + , 3 6
r × = , 2 2 a R × = . Kesik piramida · Piramidaning – H va kesik piramidaning - 1
balandliklari; l - yon qirrasi uzunligi, 1
va
2 S - piramidaning asoslari yuzi, 1
va
2 P - piramidaning asoslari piremetri, 1
va
2 a - pirami- daning asoslarnig uzunligi, hamda
-piramida, 1
hajmi bo'lsin; · Muntazam bo'lmagan kesik piramidaning yon sirti, alohida- alohida olingan yoqlari yuzining yig'indisiga teng; · Yon sirti: ( ) 1 2 1 2 yon S P P l = + × ; · To’la sirti: 1 2
yon S S S S = + + ; · Hajmi: ( ) 1 1 1 2 2 1 1 3 V S S S S H = + × + × ; · 2 1 1
H a H = , 2 2 2 1 2 1 1 S H a S H a æ ö æ ö = = ç ÷ ç ÷ è ø è ø , 2 1 1
H V H æ ö = ç ÷ è ø . Muntazam to'rtburchakli kesik piramida · Yon sirti: 1 2
( ) 2 yon S P P h = + × ; 1 4 P a = , 2 4
b = ; Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 104 · To’la sirti: 1 2
to la asos asos yon S S S S = + + ; · Hajmi: ( ) 1 1 2 2 1 1 3 V S S S S H = + × + × ; · , AN y NC x = = ; · 1 1 2 2 A C d b = = , 1 2 AC d a = = , · 1 d AC = , 2 2
y H = + · ( ) 2 2 a b x - = ; ( ) 2 2 a b y + = ; · 2 2 2 2 H d y l x = - = - . Oktaedr a - har bir qirrasi uzunligi. · To’la sirti: 2 2 3 t S a = ; · Hajmi: 3 2 3 a V = . 2 2
R = ; 6 6
r = . Dodekaedr Barcha
12 ta yoqlari muntazam beshburchakdan iborat ko’pyoq dodekaendr deyiladi. · To’la sirti: ( )
3 5 5 2 5
t S a = + ; Hajmi: ( ) 3 15 7 5 4 a V + = ; · ( ) 3 1
5 4
R + = , ( ) 1 0 2 5 1 1 5
2 0 a r + = . Ikosaedr Barcha
20 ta yoqlari teng tomonli uchburchakdan iborat ko’pyoq ikosaedr deyiladi. · To’la sirti: 2 5
t S a = ; Hajmi: ( ) 3 5 3 5 12
V + = ; · ( ) 2 5
5 4
R + = , ( ) 3 3 5 12 a r + = . Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m Click here to buy A B B Y Y PD F Transfo rm er 2 .0 w w w .A B B Y Y. c o m 105 Ko'pyoqqa ichki chizilgan shar ·
- ko'pyoq hajmi: 1 3 t V S r = × × ;
· t S - ko'pyoq to'la sirti: 3
= ; · r -ichki chizilgan shar radiusi: 3
= . Silindr R - silindr asosining radiusi, H – balandligi. ·Yon sirti: 2
p = ; ·To’la sirti: 2 (
t S R R H p = × + ; · Hajmi: 2
R H p = ; · Diagonal yoki o’q kesim yuzi: ' 2
S R H = ; · 2 2 2 2 , 4
asos S R a R H p = = + ; Silindrning yoyilmasi Konus R - konus asosining radiusi, L - yasovchisi, H - balandligi, j - yasovchi va asos tekisligi orasidagi burchak. · Yon sirti: yon S R L p = , 2
yon S R S cos cos p j j = = ; · To’la sirti: ( )
S R R L p = × + ; · Hajmi: 2 3 l V R H p = × ; · Konusga ichki chizilgan shar radiusi r ga teng bo'lsa: R H r l R × = + ; · 2 3
sil kon kon t sil V S S V = Þ = ,
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t kon sil kon t S V V S = Þ = ; · 2 2
l R H = + ; · Konus yoyilmasida radiusi l ga, yoy uzunligi 2πR ga teng
bo'lgan sektor hosil bo'ladi; · Yoyilmasi uchidagi burchak uchun: 360 R
a= o Kesik konus Kesik konus asoslarining radiuslari R va
, balandligi H bo'lsin. · Yon sirti: ( ) yon S L R r p = × + ; · To’la sirti: 2 2 ( )
S L R r R r p p p = × + + + ; · Hajmi: 2 2 1 ( ) 3 V H R R r r p = + × + ; · Konus uchidan H 1 masofada konusni S 1 yuzali doira bo'ylab kesuvchi tekislik undan
hajmli konus ajratsin, konus hajmi V , S . U holda 2 3
1 1 1 1 ; ; S H H V H r S H H R V H æ ö æ ö = = = ç ÷ ç ÷ è ø è ø . Shar va sfera · Shar sirti: 2 4
R p = , 2
d p = ; · Hajmi: 3 3
R 3 6 V d p p = = ; · Shar kesimining radiusi r va shar markazidan kesimgacha masofa
uchun:
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