INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impac ac o : 8,293 Volume 14, issue 2, No embe 2025
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HOSILANI GEOMETRIK MASALALARNI YECHISHDA TADBIQI
Umi o Bax iddin Xay illaye ich,
Xay ullaye Dilshod Bax i dino ich
Shah isabz “Temu bekla mak abi” ha biy akademik
li seyi ma ema ika ani o‘qi u chila i
Anno a siya: Hosila ma zusi o‘ a mak ab a akademik li seyla da Algeb a anidan o‘ iladi.
Lekin geome iya anida gapi ilmaydi. Hozi gi kunda ma ema ikadan milliy se i ika , hamda
a es a siya sa olla ida geome iya anidan qo‘yilgan sa olla ni yechish masalasi hosila
ma zusida bo ib aqaladi. Shu maqsadda ushbu maqolada geome iyadan masalala yechishda
hosilani adbiqiga doi masalala yechib ko‘ sa ilgan.
Kali so‘zla : Hosila, g a ik, masala, yechim, eng ka a, hajim.
Hosilani hayo da adbiqi juda ko‘p uch aydi. Geome ik jismla ni o‘lchamla ini
yasamasdan oldin hosila yo damida aniqlanadi. Shuning uchun geome iya anidan masalala ni
yechishda hosiladan oydalanamiz.
Masala-1. Tomonla i 21 a 16 ga eng bo‘lgan o‘g‘ i o‘ bu chak shaklidagi unkaning o‘ a
uchidan omoni h ga eng bo‘lgan k ad a qi qib olindi qu i shakliga kel i ildi. Qu ining hajmi
eng ka a bo‘ladigan h ni oping.
Be ilgan:
AB=21
BC=16
AK=KM=h
KN=16−2h
FF1=16−2h
h−?
Yechish: Shakil o‘g‘ i bu chakli pa allelepiped shakliga keladi eng ka a hajimga ega bo‘lish
uchun quyidagi o mulani olamiz: Vh=21−2h 16−2hh
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impac ac o : 8,293 Volume 14, issue 2, No embe 2025
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Index: google schola , esea ch ga e, esea ch bib, zenodo, open ai e.
h ps://schola .google.com/schola ?hl= u&as_sd =0%2C5&q=wosjou nals.com&b nG
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47
bu i odani soddalash i sak Vh=336h−74h2+4h3
ko‘ inishga keladi. I odadan hosila olamiz a V'h=0 englamani yechamiz.
V'h=336−148h+12h2
12h2−148h+336=0 :4
3h2−74h+84=0
bundan h1=3 a h2=91
3yechimla chiqadi.
Masala sha iga ko‘ a h=3bo‘lganda eng ka a hajimga ega bo‘ladi.
Masala-2. Radiusi Rga eng bo‘lgan ya im aylanaga CDEF o‘g‘ i o‘ bu chak ichki chizilgan
SCDEFning eng ka a qiyma ini oping.
Be ilgan:
R
SCDEFmax−?
Yechish: EF=b,OF=a
2∆OFE dan R2=b2+a2
4⟺ 4R2=4b2+a2 englikdan a=2 R2−b2
S=ab=2b R2−b2=2 b2R2−b4
Sb=2 b2R2−b4i odadan hosila olamiz.
S'b=2bR2−4b3
b2R2−b4
S'b=0deb yechamiz 2bR2−4b3=0 a b2R2−b4≠0 englikdan b=R2
a=2 R2−R2
2=2R
SCDEF= 2R∙ R2=R2
Ja ob: R2
Masala-3. Radiusi 6 2ga eng bo‘lgan sha ga ichki chizilgan eng ka a hajmli konus asosining
adiusini oping.
Be ilgan:
AO=62
AD=R−?
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impac ac o : 8,293 Volume 14, issue 2, No embe 2025
h ps://wo dlyknowledge.uz/index.php/IJSR wo ldly knowledge
Index: google schola , esea ch ga e, esea ch bib, zenodo, open ai e.
h ps://schola .google.com/schola ?hl= u&as_sd =0%2C5&q=wosjou nals.com&b nG
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48
Yechish: ∆AODdan AO2=AD2+OD2
6 2 2=R2+H−6 2 2
72=R2+H2−12 2H+72
12 2H−H2=R2
Vk=1
3πR2H=1
3π12 2H−H2H=4 2πH2−1
3πH3
Vk=4 2πH2−1
3πH3i odadan hosila olamiz.
V'H=8 2πH−πH2=πH8 2−H
V'H=0danπH≠0 aH=8 2
R2=12 2∙8 2−128=12∙16−128=64
R2=64⟺R=8
Ja ob:8.
Masala-4. Sha ga eng ka a hajmli silind ichki chizilgan R−sha adiusi, −silind asosi
adiusi.
a)
Rnisba ni oping.
b) Sha hajmining silind hajmiga nisba ini
oping.
Be ilgan:
AO=R−sha adiusi
AE= −silind asosi adiusi.
OE=H
2
R−?,Vsh
Vsil−?
Yechish: ∆AEO
danR2= 2+H2
4⟺4R2=4 2+H2⟺4R2−H2=4 2
2=R2−1
4H21
Vsil=π 2H=π R2−1
4H2H=πR2H−πH3
4
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impac ac o : 8,293 Volume 14, issue 2, No embe 2025
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49
Vsil=πR2H−πH3
4i odadan hosila olamiz
Vsil
'H= πR2H−πH3
4'=πR2−3π
4H2
Vsil
'H=0 englikdan πR2−3π
4H2=0⇔H=2R
3(2)
(2) i odani (1) i odaga qo‘yamiz
2=R2−R2
3=2R2
3⇔ = 2R
3=6R
3
a)
R=6R
3:R= 6
3
b) Vsh=4
3πR3Vsil=π 2H=43πR3
9
Vsh
Vsil=4
3πR3:4 3πR3
9=4
3πR3∙9
43πR3=33= 3
Ja ob: a) 6
3b) 3
Masala-5. Apo emasi 3 3 ga eng bo‘lgan mun azam o‘ bu chakli pi amida hajmining eng
ka a qiyma ini oping.
Be ilgan:
SABCD−mun azam pi amida
SE=3 3
Vmax−?
Yechish: ∆SEOdan SE2=SO2+OE2⇔3 3 2==H2+a2
4
108=4H2+a2⇔a2=108−4H2
VpH=1
3a2H=1
3108−4H2H=36H−4
3H3
VpH=36H−4
3H3i odadan hosila olamiz.
Vp
'H=36−4H2 englikdan 36−4H2=0 englamadan H2=9⇔H=3
a2=108−36=72
Vp=1
3a2H=1
3∙72∙3=72
Ja ob: Vmax=72
Xulosa: Geome iya ani o‘sib kelayo gan yosh a lodni kamol op i ishda o‘qu ani
si a ida keng imkoniya la ga ega. U o‘qu chi a akku ini i ojlan i ib, ula ni aqlini peshlaydi,
uni a ibga soladi, o‘qu chila ga maqsadga yo‘nal i ilganlik, man iqiy iki lash opqi lik
INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCHERS
ISSN: 3030-332X Impac ac o : 8,293 Volume 14, issue 2, No embe 2025
h ps://wo dlyknowledge.uz/index.php/IJSR wo ldly knowledge
Index: google schola , esea ch ga e, esea ch bib, zenodo, open ai e.
h ps://schola .google.com/schola ?hl= u&as_sd =0%2C5&q=wosjou nals.com&b nG
h ps://www. esea chga e.ne /p o ile/Wo ldly-Knowledge
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50
xisla la ini shakillan i ib bo adi. Shunday ekan masalala ma ema ika mu axassisligida a’lim
olayo gan alabala a o‘qu chila ning hosila yo damida geome ik masalala ni yechishda
qo‘yiladigan alabla ni oson o‘ ganishga yo dam be adi.
Foydalanilgan adabiyo la :
1. I.Is oilo , Z.Pashaye “Geome iya” Toshken -2010.
2. Sh.Jumaye , B.Ro‘ziye a boshqala “Geome iya” Toshken -2024.
3. 2025-yil BMBA sa olla o‘plami.
