Prizma

Bu nima va nega kerak

G'isht, qadoqlash qutisi, xona yoki me'moriy ustunlar geometriyada prizma deb ataluvchi ko'pyoq shakliga ega.

Xona hajmini, idish sig'imini yoki buyumning sirt yuzini hisoblashda prizma formulalaridan foydalanamiz. Ushbu darsda prizma turlari, kesimlari hamda sirt yuzi va hajmini hisoblash qoidalarini o'rganamiz.

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Asosiy g'oya va ta'riflar

1. Prizma va uning elementlari

Ikki yog'i parallel tekisliklarda yotuvchi teng ko'pburchaklar (asoslar), qolgan yoqlari esa parallelogrammlar (yon yoqlar) bo'lgan ko'pyoqqa prizma deyiladi.

       D'----------C'
      /|          /|
     / |         / |
    A'----------B' |
    |  |        |  |   H (balandlik)
    |  D--------|--C
    | /         | /
    |/          |/
    A-----------B

• Asoslari: Parallel tekisliklardagi teng ko'pburchaklar ($ABCD$ va $A'B'C'D'$).
• Yon qirralari: Yon yoqlarning kesishish kesmalari ($AA', BB', CC', DD'$). Ular o'zaro teng va parallel.
• Balandlik ($H$): Asos tekisliklari orasidagi masofa.

2. Prizma turlari

1. To'g'ri prizma: Yon qirralari asos tekisligiga perpendikulyar prizma. Yon yoqlari to'g'ri to'rtburchaklardan iborat, balandlik yon qirraga teng ($H = l$).
2. Og'ma prizma: Yon qirralari asos tekisligiga qiya joylashgan prizma ($H < l$).
3. Muntazam prizma: Asosi muntazam ko'pburchak bo'lgan to'g'ri prizma.

3. Parallelepiped va kub

Asosi parallelogramm bo'lgan prizma parallelepiped deyiladi.

• To'g'ri to'rtburchakli parallelepiped: Asosi to'g'ri to'rtburchak bo'lgan to'g'ri prizma. O'lchamlari $a, b, c$ bo'lsa, bosh diagonali: $d = \sqrt{a^2 + b^2 + c^2}$
• Kub: Barcha qirralari teng to'g'ri to'rtburchakli parallelepiped ($a = b = c$). Bosh diagonali: $d = a\sqrt{3}$.

4. Sirt yuzi va hajm formulalari

Har qanday prizma uchun to'la sirt yuzi: $S_{tola} = S_{yon} + 2S_{asos}$

To'g'ri prizma:

• $S_{yon} = P_{asos} \cdot H$ ($P_{asos}$ — asos perimetri).
• $V = S_{asos} \cdot H$.

To'g'ri to'rtburchakli parallelepiped ($a, b, c$):

• $S_{tola} = 2(ab + bc + ac)$
• $V = a \cdot b \cdot c$

Kub ($a$):

• $S_{tola} = 6a^2$
• $V = a^3$

Og'ma prizma:

• $V = S_{asos} \cdot H = S_{kesim} \cdot l$
• $S_{yon} = P_{kesim} \cdot l$ (bu yerda $P_{kesim}$ va $S_{kesim}$ — perpendikulyar kesim parametrlari).

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Yechilgan misollar

1-misol. To'g'ri to'rtburchakli parallelepiped

O'lchamlari $3\text{ sm}$, $4\text{ sm}$ va $12\text{ sm}$ bo'lgan to'g'ri to'rtburchakli parallelepipedning bosh diagonali, to'la sirti va hajmini toping.

• Bosh diagonal: $d = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13\text{ sm}$
• To'la sirti: $S_{tola} = 2(3 \cdot 4 + 4 \cdot 12 + 3 \cdot 12) = 2(12 + 48 + 36) = 192\text{ sm}^2$
• Hajmi: $V = 3 \cdot 4 \cdot 12 = 144\text{ sm}^3$

Javob: $d = 13\text{ sm}$, $S_{tola} = 192\text{ sm}^2$, $V = 144\text{ sm}^3$.

2-misol. Muntazam uchburchakli prizma

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

• To'g'ri prizmada $H = l = 6\text{ sm}$.
• Asos perimetri va yuzi: $P_{asos} = 3 \cdot 4 = 12\text{ sm}, \quad S_{asos} = \frac{4^2\sqrt{3}}{4} = 4\sqrt{3}\text{ sm}^2$
• Yon sirti: $S_{yon} = 12 \cdot 6 = 72\text{ sm}^2$
• To'la sirti: $S_{tola} = 72 + 2 \cdot 4\sqrt{3} = 72 + 8\sqrt{3}\text{ sm}^2$
• Hajmi: $V = 4\sqrt{3} \cdot 6 = 24\sqrt{3}\text{ sm}^3$

Javob: $S_{yon} = 72\text{ sm}^2$, $S_{tola} = 72 + 8\sqrt{3}\text{ sm}^2$, $V = 24\sqrt{3}\text{ sm}^3$.

3-misol. To'g'ri burchakli uchburchak asosli prizma

To'g'ri prizmaning asosi katetlari $6\text{ sm}$ va $8\text{ sm}$ bo'lgan to'g'ri burchakli uchburchak. Eng katta yon yoq kvadratdan iborat. Prizmaning to'la sirti va hajmini toping.

• Gipotenuza: $c = \sqrt{6^2 + 8^2} = 10\text{ sm}$.
• Eng katta yon yoq gipotenuzaga tiralgan kvadrat, demak $H = 10\text{ sm}$.
• Asos parametrlari: $P_{asos} = 6 + 8 + 10 = 24\text{ sm}, \quad S_{asos} = \frac{6 \cdot 8}{2} = 24\text{ sm}^2$
• Sirtlari va hajmi: $S_{yon} = 24 \cdot 10 = 240\text{ sm}^2, \quad S_{tola} = 240 + 2 \cdot 24 = 288\text{ sm}^2$ $V = 24 \cdot 10 = 240\text{ sm}^3$

Javob: $S_{tola} = 288\text{ sm}^2$, $V = 240\text{ sm}^3$.

4-misol. Romb asosli to'g'ri prizma

To'g'ri prizmaning asosi diagonallari $12\text{ sm}$ va $16\text{ sm}$ bo'lgan rombdan iborat. Yon qirrasi $10\text{ sm}$ bo'lsa, to'la sirti va hajmini toping.

• Romb tomoni: $a = \sqrt{6^2 + 8^2} = 10\text{ sm}$.
• Asos yuzi: $S_{asos} = \frac{12 \cdot 16}{2} = 96\text{ sm}^2$.
• Yon sirti ($H = 10\text{ sm}$): $S_{yon} = (4 \cdot 10) \cdot 10 = 400\text{ sm}^2$.
• To'la sirti: $S_{tola} = 400 + 2 \cdot 96 = 592\text{ sm}^2$.
• Hajmi: $V = 96 \cdot 10 = 960\text{ sm}^3$.

Javob: $S_{tola} = 592\text{ sm}^2$, $V = 960\text{ sm}^3$.

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Mashq

1. To'g'ri to'rtburchakli parallelepipedning o'lchamlari $4\text{ sm}$, $6\text{ sm}$ va $12\text{ sm}$. Bosh diagonali va hajmini toping.
2. Kubning bosh diagonali $6\sqrt{3}\text{ sm}$. Uning to'la sirti va hajmini hisoblang.
3. Muntazam to'rtburchakli prizma asosining diagonali $8\sqrt{2}\text{ sm}$, balandligi $15\text{ sm}$. To'la sirtini toping.
4. To'g'ri prizmaning asosi tomonlari $9\text{ sm}$, $10\text{ sm}$ va $17\text{ sm}$ bo'lgan uchburchak, balandligi $8\text{ sm}$. Hajmini toping.
5. Muntazam uchburchakli prizmaning yon sirti $180\text{ sm}^2$, balandligi $12\text{ sm}$. Asosining tomonini toping.
6. Muntazam oltiburchakli to'g'ri prizma asosining tomoni $4\text{ sm}$, balandligi $5\text{ sm}$. Yon sirti va hajmini toping.
7. To'g'ri to'rtburchakli parallelepiped asosi tomoni $6\text{ sm}$ bo'lgan kvadrat. Bosh diagonali asos tekisligi bilan $30^\circ$ li burchak hosil qilsa, balandligi va hajmini toping.
8. To'g'ri prizmaning asosi romb bo'lib, tomoni $13\text{ sm}$, bir diagonali $10\text{ sm}$. Balandligi $7\text{ sm}$ bo'lsa, hajmini toping.

Javoblar va yechim yo'llari

1. $d = \sqrt{4^2 + 6^2 + 12^2} = 14\text{ sm}$. $V = 4 \cdot 6 \cdot 12 = 288\text{ sm}^3$. Javob: $d = 14\text{ sm}$, $V = 288\text{ sm}^3$.
2. $d = a\sqrt{3} \implies a = 6\text{ sm}$. $S_{tola} = 6 \cdot 36 = 216\text{ sm}^2$, $V = 6^3 = 216\text{ sm}^3$. Javob: $S_{tola} = 216\text{ sm}^2$, $V = 216\text{ sm}^3$.
3. Kvadrat tomoni $a = 8\text{ sm}$. $S_{asos} = 64\text{ sm}^2$. $S_{yon} = 32 \cdot 15 = 480\text{ sm}^2$. $S_{tola} = 480 + 2 \cdot 64 = 608\text{ sm}^2$. Javob: $608\text{ sm}^2$.
4. Geron: $p = 18\text{ sm}$, $S_{asos} = \sqrt{18 \cdot 9 \cdot 8 \cdot 1} = 36\text{ sm}^2$. $V = 36 \cdot 8 = 288\text{ sm}^3$. Javob: $288\text{ sm}^3$.
5. $P_{asos} = 180 / 12 = 15\text{ sm}$. Muntazam uchburchak tomoni: $a = 15 / 3 = 5\text{ sm}$. Javob: $5\text{ sm}$.
6. $P_{asos} = 24\text{ sm}$, $S_{yon} = 24 \cdot 5 = 120\text{ sm}^2$. $S_{asos} = 6 \cdot \frac{16\sqrt{3}}{4} = 24\sqrt{3}\text{ sm}^2$. $V = 24\sqrt{3} \cdot 5 = 120\sqrt{3}\text{ sm}^3$. Javob: $S_{yon} = 120\text{ sm}^2$, $V = 120\sqrt{3}\text{ sm}^3$.
7. Asos diagonali $6\sqrt{2}\text{ sm}$. $H = 6\sqrt{2} \cdot \operatorname{tg} 30^\circ = 2\sqrt{6}\text{ sm}$. $V = 36 \cdot 2\sqrt{6} = 72\sqrt{6}\text{ sm}^3$. Javob: $H = 2\sqrt{6}\text{ sm}$, $V = 72\sqrt{6}\text{ sm}^3$.
8. Ikkinchi diagonal: $2 \cdot \sqrt{13^2 - 5^2} = 24\text{ sm}$. $S_{asos} = \frac{10 \cdot 24}{2} = 120\text{ sm}^2$. $V = 120 \cdot 7 = 840\text{ sm}^3$. Javob: $840\text{ sm}^3$.

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Keng tarqalgan xatolar

1. To'la sirtda bitta asos yuzini qo'shish bilan cheklanish. ✗ $S_{tola} = S_{yon} + S_{asos}$ deb yozish. ✓ Prizmaning ikkita asosi bor: $S_{tola} = S_{yon} + 2S_{asos}$.

2. Og'ma prizmada yon qirrani balandlik deb hisoblash. ✗ $V = S_{asos} \cdot l$ deb hisoblash. ✓ Og'ma prizmada $H = l \cdot \sin \alpha \le l$ bo'lib, hajm $V = S_{asos} \cdot H$ dir.

3. Muntazam prizma bilan to'g'ri prizmani adashtirish. ✗ Asosi to'g'ri to'rtburchak bo'lgan to'g'ri prizmani "muntazam" deb atash. ✓ Muntazam prizma bo'lishi uchun prizma to'g'ri bo'lishi va asosi muntazam ko'pburchak bo'lishi shart.

4. Parallelepiped diagonalini bitta yoq diagonali bilan adashtirish. ✗ $d = \sqrt{a^2 + b^2}$ deb olish. ✓ Fazoviy bosh diagonal uchala qirraga bog'liq: $d = \sqrt{a^2 + b^2 + c^2}$.

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