Jismlarning hajmi va sirt yuzi

Bu nima va nega kerak

Fazoviy jismlarning sirt yuzi va hajmini hisoblash muhandislik, metallurgiya va arxitekturada eng muhim amaliy vositadir.

DTM imtihonlarida stereometriya savollari ko'pincha jismlar kombinatsiyasiga (sharga tashqi chizilgan silindr, konus ichidagi shar, kub ichidagi piramida) yoki shakl o'zgarishiga (eritib qayta quyish) asoslanadi.

Ushbu darsda ko'pyoqlar va aylanish jismlari formulalarini umumlashtirib, aralash masalalarni yechamiz.

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Asosiy g'oya va formulalar jadvali

1. Barcha fazoviy jismlarning umumiy jadvali

Jism	Asos yuzi ($S_{asos}$)	Yon sirti ($S_{yon}$)	To'la sirti ($S_{to'la}$)	Hajmi ($V$)
To'g'ri prizma	Shakliga bog'liq	$P_{asos} \cdot H$	$S_{yon} + 2S_{asos}$	$S_{asos} \cdot H$
Kub ($a$)	$a^2$	$4a^2$	$6a^2$	$a^3$
Parallelepiped ($a, b, c$)	$ab$	$2c(a + b)$	$2(ab + bc + ac)$	$abc$
Muntazam piramida	Shakliga bog'liq	$\frac{1}{2} P_{asos} \cdot h_a$	$S_{yon} + S_{asos}$	$\frac{1}{3} S_{asos} \cdot H$
Silindr	$\pi R^2$	$2\pi R H$	$2\pi R (H + R)$	$\pi R^2 H$
Konus	$\pi R^2$	$\pi R l$	$\pi R (l + R)$	$\frac{1}{3}\pi R^2 H$
Shar va sfera	—	—	$4\pi R^2$	$\frac{4}{3}\pi R^3$

2. O'xshash fazoviy jismlar xossasi

Agar ikkita fazoviy jism o'xshash bo'lib, ularning mos chiziqli o'lchovlari nisbati $k$ bo'lsa:

[!IMPORTANT]

1. Chiziqli o'lchovlar (qirra, balandlik, radius): $\dfrac{a_1}{a_2} = k$
2. Yuzalar nisbati (yon sirt, to'la sirt, asos yuzi): $\dfrac{S_1}{S_2} = k^2$
3. Hajmlar nisbati: $\dfrac{V_1}{V_2} = k^3$

3. Eng muhim jismlar kombinatsiyasi

1. Silindrga ichki chizilgan shar: $H = 2R$, $V_{shar} = \frac{2}{3} V_{sil}$, $S_{shar} = \frac{2}{3} S_{sil\_to'la}$.
2. Kub va shar: Ichki shar: $r = \frac{a}{2}$. Tashqi shar: $R = \frac{a\sqrt{3}}{2}$.
3. Konusga ichki chizilgan shar: Shar radiusi konus o'q kesimiga (teng yonli uchburchak) ichki chizilgan aylana radiusi bilan bir xil: $r = \frac{S_{\triangle}}{p_{\triangle}}$.

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Yechilgan misollar

1-misol. Shakl o'zgarishi (eritib qayta quyish)

Radiusi $R = 6\text{ sm}$ bo'lgan metall shar eritilib, asosi radiusi $r = 3\text{ sm}$ bo'lgan to'g'ri silindr quyildi. Silindrning balandligini toping.

• Shar hajmini hisoblaymiz: $V_{shar} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi \cdot 6^3 = \frac{4}{3}\pi \cdot 216 = 288\pi\text{ sm}^3$
• Silindr hajmi shar hajmiga teng: $V_{sil} = \pi r^2 H = \pi \cdot 3^2 \cdot H = 9\pi H$
• Tenglashtiramiz: $9\pi H = 288\pi \implies H = \frac{288}{9} = 32\text{ sm}$

Javob: $32\text{ sm}$.

2-misol. Konusga ichki chizilgan shar

Asosi radiusi $R = 6\text{ sm}$, balandligi $H = 8\text{ sm}$ bo'lgan konusga shar ichki chizilgan. Shar hajmini toping.

• Konus yasovchisi: $l = \sqrt{6^2 + 8^2} = 10\text{ sm}$.
• Konusning o'q kesimi asosi $2R = 12\text{ sm}$, yon tomonlari $10\text{ sm}$ bo'lgan teng yonli uchburchakdir.
• O'q kesimi yuzi va yarim perimetri: $S = \frac{1}{2} \cdot 12 \cdot 8 = 48\text{ sm}^2, \quad p = \frac{10 + 10 + 12}{2} = 16\text{ sm}$
• Ichki chizilgan shar radiusi: $r = \frac{S}{p} = \frac{48}{16} = 3\text{ sm}$
• Shar hajmi: $V_{shar} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi \cdot 3^3 = 36\pi\text{ sm}^3$

Javob: $36\pi\text{ sm}^3$.

3-misol. Kubga ichki va tashqi chizilgan sharlar nisbati

Qirrasi $a$ bo'lgan kubga ichki va tashqi chizilgan sharlar hajmlari nisbatini toping.

• Ichki chizilgan shar radiusi: $r = \frac{a}{2}$.
• Tashqi chizilgan shar radiusi: $R = \frac{a\sqrt{3}}{2}$.
• Radiuslar nisbati: $\frac{R}{r} = \frac{\frac{a\sqrt{3}}{2}}{\frac{a}{2}} = \sqrt{3}$
• Hajmlar nisbati radiuslar nisbatining kubiga teng: $\frac{V_{tashqi}}{V_{ichki}} = \left(\frac{R}{r}\right)^3 = (\sqrt{3})^3 = 3\sqrt{3}$

Javob: $3\sqrt{3}$.

4-misol. Muntazam oltiburchakli prizma

Asosining tomoni $4\text{ sm}$, balandligi $6\text{ sm}$ bo'lgan muntazam oltiburchakli prizmaning to'la sirti va hajmini toping.

• Muntazam oltiburchak yuzi 6 ta muntazam uchburchak yuziga teng: $S_{asos} = 6 \cdot \frac{a^2\sqrt{3}}{4} = 6 \cdot \frac{16\sqrt{3}}{4} = 24\sqrt{3}\text{ sm}^2$
• Prizma hajmi: $V = S_{asos} \cdot H = 24\sqrt{3} \cdot 6 = 144\sqrt{3}\text{ sm}^3$
• Yon sirti va to'la sirti: $S_{yon} = P_{asos} \cdot H = (6 \cdot 4) \cdot 6 = 144\text{ sm}^2$ $S_{to'la} = S_{yon} + 2S_{asos} = 144 + 48\sqrt{3}\text{ sm}^2$

Javob: $V = 144\sqrt{3}\text{ sm}^3$, $S_{to'la} = 144 + 48\sqrt{3}\text{ sm}^2$.

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Mashq

1. Kubning fazoviy diagonali $6\sqrt{3}\text{ sm}$. Uning to'la sirti va hajmini toping.
2. To'g'ri to'rtburchakli parallelepiped o'lchovlari $3\text{ sm}, 4\text{ sm}, 12\text{ sm}$. Uning fazoviy diagonali va to'la sirtini hisoblang.
3. Muntazam uchburchakli piramidaning asos tomoni $6\text{ sm}$, balandligi $4\text{ sm}$. Piramida hajmini toping.
4. Silindr va konusning asoslari va hajmlari teng. Konus balandligi $18\text{ sm}$ bo'lsa, silindr balandligini toping.
5. Sharga tashqi chizilgan silindr hajmi $54\pi\text{ sm}^3$. Shar hajmini toping.
6. Muntazam to'rtburchakli prizma asosi tomoni $5\text{ sm}$, to'la sirti $150\text{ sm}^2$. Uning hajmini toping.
7. Radiusi $4\text{ sm}$ bo'lgan shar eritilib, asosi radiusi $4\text{ sm}$ bo'lgan konus quyildi. Konus balandligini toping.
8. Ikkita o'xshash piramidaning hajmlari nisbati $1 : 27$. Ularning yon sirtlari yuzalari nisbatini hisoblang.
9. Asos radiusi $8\text{ sm}$, balandligi $6\text{ sm}$ bo'lgan konusga ichki chizilgan shar radiusini toping.

Javoblar va yechim yo'llari

1. $d = a\sqrt{3} = 6\sqrt{3} \implies a = 6\text{ sm}$. $S_{to'la} = 6a^2 = 216\text{ sm}^2$, $V = a^3 = 216\text{ sm}^3$. Javob: $216\text{ sm}^2$, $216\text{ sm}^3$.
2. $d = \sqrt{3^2 + 4^2 + 12^2} = 13\text{ sm}$. $S_{to'la} = 2(3 \cdot 4 + 4 \cdot 12 + 3 \cdot 12) = 192\text{ sm}^2$. Javob: $13\text{ sm}$, $192\text{ sm}^2$.
3. $S_{asos} = \frac{6^2\sqrt{3}}{4} = 9\sqrt{3}\text{ sm}^2$. $V = \frac{1}{3} \cdot 9\sqrt{3} \cdot 4 = 12\sqrt{3}\text{ sm}^3$. Javob: $12\sqrt{3}\text{ sm}^3$.
4. $\pi R^2 H_{sil} = \frac{1}{3}\pi R^2 H_{kon} \implies H_{sil} = \frac{18}{3} = 6\text{ sm}$. Javob: $6\text{ sm}$.
5. $V_{shar} = \frac{2}{3} V_{sil} = \frac{2}{3} \cdot 54\pi = 36\pi\text{ sm}^3$. Javob: $36\pi\text{ sm}^3$.
6. $S_{asos} = 25\text{ sm}^2$. $S_{yon} = 150 - 2 \cdot 25 = 100\text{ sm}^2$. $P_{asos} = 20\text{ sm} \implies H = 100 / 20 = 5\text{ sm}$. $V = 25 \cdot 5 = 125\text{ sm}^3$. Javob: $125\text{ sm}^3$.
7. $V_{shar} = \frac{4}{3}\pi \cdot 64 = \frac{256\pi}{3}$. $V_{kon} = \frac{1}{3}\pi \cdot 16 \cdot H = \frac{16\pi H}{3}$. Tenglashtirsak: $16H = 256 \implies H = 16\text{ sm}$. Javob: $16\text{ sm}$.
8. $k^3 = \frac{1}{27} \implies k = \frac{1}{3}$. Sirtlar nisbati $k^2 = \frac{1}{9}$. Javob: $1 : 9$.
9. Yasovchi $l = \sqrt{8^2 + 6^2} = 10\text{ sm}$. O'q kesimi asosi $16\text{ sm}$, yon tomonlari $10\text{ sm}$. $S = \frac{1}{2} \cdot 16 \cdot 6 = 48\text{ sm}^2$, $p = \frac{10 + 10 + 16}{2} = 18\text{ sm}$. $r = \frac{48}{18} = \frac{8}{3}\text{ sm}$. Javob: $\frac{8}{3}\text{ sm}$.

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Keng tarqalgan xatolar

1. O'xshash jismlarda hajm nisbatini kvadrat deb olish. ✗ Hajmlar nisbatini $k^2$ ga teng deb hisoblash. ✓ Yuzalar nisbati $k^2$, hajmlar nisbati esa har doim $k^3$ ga teng bo'ladi.

2. Eritib qayta quyishda sirt yuzasi saqlanadi deb o'ylash. ✗ Yangi jismning sirtini eski jism sirtiga tenglashtirish. ✓ Modda qayta quyilganda uning massasi va hajmi saqlanadi, sirt yuzasi esa o'zgaradi.

3. Sharga tashqi chizilgan silindrda balandlikni radiusga teng deb olish. ✗ $H = R$ deb xato qilish. ✓ Silindr sharni to'la qamrab olgani uchun uning balandligi shar diametriga teng: $H = 2R$.

4. To'la sirtda asoslar sonini unutish. ✗ Prizma va silindrda bitta asos yuzini qo'shib qo'yish. ✓ Prizma va silindrda ikkita asos mavjud ($2S_{asos}$), piramida va konusda esa bitta asos bo'ladi.

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DTM uslubidagi test

Yuqoridagi frontmatter bo'limida keltirilgan 5 ta test savolini ishlab, javoblaringizni tekshiring.

Keyingi qadam

Siz 11-sinf geometriya kursining eng asosiy bo'limi — Fazoviy jismlar mavzularini to'liq o'rganib chiqdingiz. Endi stereometriya bo'yicha olgan bilimlaringizni DTM blok testlarida va aralash variantlarda qo'llab, mustahkamlashingiz mumkin.