Piramida

Bu nima va nega kerak

Qadimgi Misr ehromlari va zamonaviy binolarning qiya tomlari geometriyada piramida deb ataluvchi shaklga ega.

Prizmadan farqli ravishda, piramidaning bitta asosi bo'lib, uning yon yoqlari umumiy cho'qqida tutashadi. Arxitektura va qurilishda qiya tomlar yuzasi hamda fazoviy sig'imni aniqlashda piramida formulalari qo'llanadi. Ushbu darsda muntazam piramida, uning apofemasi, sirt yuzi va hajmini hisoblashni o'rganamiz.

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Asosiy g'oya va ta'riflar

1. Piramida elementlari

Asosi ko'pburchak, qolgan yoqlari umumiy bitta uchga ega bo'lgan uchburchaklardan iborat ko'pyoqqa piramida deyiladi.

            S (cho'qqi)
           /|\
          / | \
         /  |  \
        /   |H  \
       /    |    \
      D-----|-----C
     /      |    /
    /       O   /
   A-----------B

• Asosi: Ko'pburchak ($ABCD$).
• Uchi (cho'qqisi): Yon yoqlarning umumiy uchi ($S$).
• Yon qirralari: Cho'qqini asos uchlari bilan tutashtiruvchi kesmalar ($SA, SB, SC, SD$).
• Balandligi ($H$): Cho'qqidan asos tekisligiga tushirilgan perpendikulyar ($SO$).

2. Muntazam piramida va apofema

Asosi muntazam ko'pburchak bo'lib, cho'qqisining asos tekisligidagi proyeksiyasi asos markaziga tushuvchi piramida muntazam piramida deyiladi.

• Barcha yon qirralari tengdir: $SA = SB = SC = SD = b$.
• Barcha yon yoqlari teng teng yonli uchburchaklardir.
• Apofema ($h_a$): Cho'qqidan yon yoq asosiga tushirilgan balandlik.

3. Asosiy to'g'ri burchakli uchburchaklar

1. Balandlik, apofema va ichki aylana radiusi: $h_a^2 = H^2 + r^2$
2. Balandlik, yon qirra va tashqi aylana radiusi: $b^2 = H^2 + R^2$
3. Apofema, asos tomonining yarmi va yon qirra: $b^2 = h_a^2 + \left(\frac{a}{2}\right)^2$

4. Sirt yuzi va hajm formulalari

Har qanday piramida uchun: $S_{tola} = S_{yon} + S_{asos}$ $V = \frac{1}{3} S_{asos} \cdot H$

Muntazam piramida yon sirti: $S_{yon} = \frac{1}{2} P_{asos} \cdot h_a$ Barcha yon yoqlar asos bilan $\alpha$ burchak hosil qilsa: $S_{yon} = \frac{S_{asos}}{\cos \alpha}$.

Muntazam tetraedr ($a$ qirrali): $S_{tola} = a^2\sqrt{3}, \quad V = \frac{a^3\sqrt{2}}{12}$

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Yechilgan misollar

1-misol. Muntazam to'rtburchakli piramida

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

• Asos kvadrat: $r = \frac{a}{2} = 3\text{ sm}$.
• Apofema: $h_a = \sqrt{H^2 + r^2} = \sqrt{4^2 + 3^2} = 5\text{ sm}$
• Asos yuzi va perimetri: $S_{asos} = 6^2 = 36\text{ sm}^2, \quad P_{asos} = 4 \cdot 6 = 24\text{ sm}$
• Sirtlari va hajmi: $S_{yon} = \frac{1}{2} \cdot 24 \cdot 5 = 60\text{ sm}^2, \quad S_{tola} = 60 + 36 = 96\text{ sm}^2$ $V = \frac{1}{3} \cdot 36 \cdot 4 = 48\text{ sm}^3$

Javob: $h_a = 5\text{ sm}$, $S_{tola} = 96\text{ sm}^2$, $V = 48\text{ sm}^3$.

2-misol. Muntazam uchburchakli piramida

Asosining tomoni $6\sqrt{3}\text{ sm}$, yon qirrasi $10\text{ sm}$ bo'lgan muntazam uchburchakli piramidaning balandligi va hajmini toping.

• Tashqi aylana radiusi: $R = \frac{a}{\sqrt{3}} = \frac{6\sqrt{3}}{\sqrt{3}} = 6\text{ sm}$.
• Balandlik: $H = \sqrt{b^2 - R^2} = \sqrt{10^2 - 6^2} = 8\text{ sm}$
• Asos yuzi: $S_{asos} = \frac{(6\sqrt{3})^2\sqrt{3}}{4} = 27\sqrt{3}\text{ sm}^2$
• Hajmi: $V = \frac{1}{3} \cdot 27\sqrt{3} \cdot 8 = 72\sqrt{3}\text{ sm}^3$

Javob: $H = 8\text{ sm}$, $V = 72\sqrt{3}\text{ sm}^3$.

3-misol. Yon qirralari teng burchak ostida og'gan piramida

Asosi katetlari $12\text{ sm}$ va $16\text{ sm}$ bo'lgan to'g'ri burchakli uchburchak. Barcha yon qirralar asos bilan $45^\circ$ li burchak hosil qilsa, balandligi va hajmini toping.

• Qoida: Barcha yon qirralar asos bilan bir xil burchak hosil qilsa, cho'qqi tashqi aylana markaziga tushadi.
• Gipotenuza: $c = \sqrt{12^2 + 16^2} = 20\text{ sm} \implies R = \frac{c}{2} = 10\text{ sm}$.
• Balandlik: $H = R \cdot \operatorname{tg} 45^\circ = 10 \cdot 1 = 10\text{ sm}$.
• Asos yuzi va hajmi: $S_{asos} = \frac{12 \cdot 16}{2} = 96\text{ sm}^2, \quad V = \frac{1}{3} \cdot 96 \cdot 10 = 320\text{ sm}^3$

Javob: $H = 10\text{ sm}$, $V = 320\text{ sm}^3$.

4-misol. Yon yoqlari teng burchak ostida og'gan piramida

Piramidaning asosi tomonlari $13\text{ sm}$, $14\text{ sm}$ va $15\text{ sm}$ bo'lgan uchburchak. Barcha yon yoqlar asos bilan $60^\circ$ li burchak hosil qiladi. Balandligi va to'la sirtini toping.

• Qoida: Yon yoqlar teng burchak hosil qilsa, cho'qqi ichki markazga tushadi va $S_{yon} = \frac{S_{asos}}{\cos \alpha}$.
• Geron formulasi: $p = 21\text{ sm}$, $S_{asos} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84\text{ sm}^2$.
• Ichki radius: $r = \frac{84}{21} = 4\text{ sm}$.
• Balandlik: $H = r \cdot \operatorname{tg} 60^\circ = 4\sqrt{3}\text{ sm}$.
• Sirtlar: $S_{yon} = \frac{84}{\cos 60^\circ} = \frac{84}{0{,}5} = 168\text{ sm}^2, \quad S_{tola} = 168 + 84 = 252\text{ sm}^2$

Javob: $H = 4\sqrt{3}\text{ sm}$, $S_{tola} = 252\text{ sm}^2$.

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Mashq

1. Muntazam to'rtburchakli piramidaning asos tomoni $8\text{ sm}$, balandligi $3\text{ sm}$. Apofemasi, to'la sirti va hajmini toping.
2. Muntazam to'rtburchakli piramidaning asos tomoni $8\text{ sm}$, apofemasi $5\text{ sm}$. Balandligi va hajmini toping.
3. Muntazam uchburchakli piramidaning asos tomoni $6\sqrt{3}\text{ sm}$, balandligi $4\text{ sm}$. Apofemasi va yon sirtini toping.
4. Barcha qirralari $4\text{ sm}$ bo'lgan muntazam tetraedrning to'la sirti va hajmini toping.
5. Piramidaning asosi tomonlari $6\text{ sm}$ va $8\text{ sm}$ bo'lgan to'g'ri to'rtburchak. Cho'qqi diagonallar kesishish nuqtasiga tushadi, balandligi $12\text{ sm}$. Yon qirrasini toping.
6. Piramida asosi yuzi $36\text{ sm}^2$, balandligi $6\text{ sm}$. Asosga parallel tekislik balandlikni cho'qqidan $1 : 2$ nisbatda bo'lsa, kesim yuzini toping.
7. Muntazam to'rtburchakli piramidaning to'la sirti $96\text{ sm}^2$, yon sirti $60\text{ sm}^2$. Asosining tomonini va balandligini toping.
8. Kesik piramida asoslari yuzlari $9\text{ sm}^2$ va $25\text{ sm}^2$, balandligi $6\text{ sm}$. Hajmini toping ($V = \frac{1}{3}H(S_1 + \sqrt{S_1 S_2} + S_2)$).

Javoblar va yechim yo'llari

1. $r = 4\text{ sm}$, $h_a = \sqrt{3^2 + 4^2} = 5\text{ sm}$. $S_{yon} = \frac{1}{2} \cdot 32 \cdot 5 = 80\text{ sm}^2$. $S_{tola} = 80 + 64 = 144\text{ sm}^2$. $V = \frac{1}{3} \cdot 64 \cdot 3 = 64\text{ sm}^3$. Javob: $h_a = 5\text{ sm}$, $S_{tola} = 144\text{ sm}^2$, $V = 64\text{ sm}^3$.
2. $r = 4\text{ sm}$. $H = \sqrt{5^2 - 4^2} = 3\text{ sm}$. $V = \frac{1}{3} \cdot 64 \cdot 3 = 64\text{ sm}^3$. Javob: $H = 3\text{ sm}$, $V = 64\text{ sm}^3$.
3. $r = \frac{6\sqrt{3}}{2\sqrt{3}} = 3\text{ sm}$. $h_a = \sqrt{4^2 + 3^2} = 5\text{ sm}$. $S_{yon} = \frac{1}{2} \cdot 18\sqrt{3} \cdot 5 = 45\sqrt{3}\text{ sm}^2$. Javob: $h_a = 5\text{ sm}$, $S_{yon} = 45\sqrt{3}\text{ sm}^2$.
4. $S_{tola} = 4^2\sqrt{3} = 16\sqrt{3}\text{ sm}^2$. $V = \frac{4^3\sqrt{2}}{12} = \frac{16\sqrt{2}}{3}\text{ sm}^3$. Javob: $S_{tola} = 16\sqrt{3}\text{ sm}^2$, $V = \frac{16\sqrt{2}}{3}\text{ sm}^3$.
5. $R = \frac{\sqrt{6^2 + 8^2}}{2} = 5\text{ sm}$. $b = \sqrt{12^2 + 5^2} = 13\text{ sm}$. Javob: $13\text{ sm}$.
6. Cho'qqidan masofa nisbati $k = \frac{1}{3}$. $S_{kesim} = 36 \cdot \frac{1}{9} = 4\text{ sm}^2$. Javob: $4\text{ sm}^2$.
7. $S_{asos} = 96 - 60 = 36\text{ sm}^2 \implies a = 6\text{ sm}$. $P_{asos} = 24\text{ sm}$, $h_a = 5\text{ sm}$. $r = 3\text{ sm}$, $H = \sqrt{5^2 - 3^2} = 4\text{ sm}$. Javob: $a = 6\text{ sm}$, $H = 4\text{ sm}$.
8. $V = \frac{1}{3} \cdot 6 \cdot (9 + 15 + 25) = 2 \cdot 49 = 98\text{ sm}^3$. Javob: $98\text{ sm}^3$.

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Keng tarqalgan xatolar

1. Apofema bilan balandlikni adashtirish. ✗ $S_{yon} = \frac{1}{2} P_{asos} \cdot H$ deb hisoblash. ✓ Yon sirt apofemaga asoslanadi: $S_{yon} = \frac{1}{2} P_{asos} \cdot h_a$.

2. Hajm formulasida $\frac{1}{3}$ koeffitsiyentini unutish. ✗ $V = S_{asos} \cdot H$ deb prizma kabi hisoblash. ✓ Piramida hajmi: $V = \frac{1}{3} S_{asos} \cdot H$.

3. Ichki va tashqi aylanalar radiuslarini aralashtirish. ✗ Apofemani topishda $R$ dan, yon qirrada $r$ dan foydalanish. ✓ Apofema $r$ bilan ($h_a^2 = H^2 + r^2$), yon qirra esa $R$ bilan ($b^2 = H^2 + R^2$) bog'liq.

4. To'la sirtda ikkita asos yuzini qo'shib qo'yish. ✗ $S_{tola} = S_{yon} + 2S_{asos}$ deb yozish. ✓ Piramidada bitta asos mavjud: $S_{tola} = S_{yon} + S_{asos}$.

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