Muntazam ko'pburchaklar

Bu nima va nega kerak

Tabiatda asalarilar uya qurayotganda eng tejamkor va mustahkam geometrik shakl — muntazam oltiburchaklardan foydalanadi. Buning sababi oddiy: tekislikni oraliqsiz to'ldirish (koshin terish) mumkin bo'lgan muntazam ko'pburchaklar orasida oltiburchak eng kam perimetr bilan eng katta yuzani o'rab oladi.

Texnikada bolt va gaykalar, me'morchilikdagi gumbaz va ustunlar ham ko'pincha muntazam ko'pburchak shaklida ishlanadi.

Tomonlari va burchaklari teng bo'lgan bunday shakllar aylanani teng bo'laklarga bo'lish orqali hosil qilinadi. Ushbu darsda muntazam ko'pburchaklarning burchaklari, yuzasi hamda ularga ichki va tashqi chizilgan aylanalar orasidagi bog'liqlikni o'rganamiz.

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Asosiy g'oya va ta'riflar

[!IMPORTANT] Ta'rif. Barcha tomonlari va barcha burchaklari teng bo'lgan qavariq ko'pburchak muntazam ko'pburchak deyiladi.

Masalan:

• $n = 3$ bo'lsa — muntazam (teng tomonli) uchburchak;
• $n = 4$ bo'lsa — muntazam to'rtburchak (kvadrat);
• $n = 6$ bo'lsa — muntazam oltiburchak.

    Muntazam uchburchak       Kvadrat           Muntazam oltiburchak
           /\                 +-------+                 / \
          /  \                |       |               /     \
         /____\               +-------+               \     /
                                                        \ /

1. Burchaklar va ularning o'lchovlari

$n$ tomonli muntazam ko'pburchakda:

1. Ichki burchaklar yig'indisi: $S_n = (n - 2) \cdot 180^\circ$
2. Bitta ichki burchagi: $\alpha_n = \frac{(n - 2) \cdot 180^\circ}{n} = 180^\circ - \frac{360^\circ}{n}$
3. Bitta tashqi burchagi: $\beta_n = \frac{360^\circ}{n}$ (Har qanday qavariq ko'pburchakning tashqi burchaklari yig'indisi doimo $360^\circ$ ga teng).
4. Markaziy burchagi: $\gamma_n = \frac{360^\circ}{n}$
5. Diagonallar soni: $d = \frac{n(n - 3)}{2}$

2. Ichki va tashqi aylanalar

Har qanday muntazam ko'pburchakka ham ichki, ham tashqi aylana chizish mumkin va ularning markazlari bitta nuqtada ustma-ust tushadi.

• Tashqi aylana radiusi ($R$): markazdan uchlarigacha bo'lgan masofa.
• Ichki aylana radiusi ($r$ — apofema): markazdan tomonlarigacha bo'lgan masofa.
• Bog'lanish formulalari: $a_n = 2R \sin \frac{180^\circ}{n}, \quad r = R \cos \frac{180^\circ}{n}, \quad r = \frac{a_n}{2 \operatorname{tg} \frac{180^\circ}{n}}$
• Yuza formulasi: $S = \frac{1}{2} P \cdot r = \frac{n}{2} a_n \cdot r$

3. Eng muhim muntazam ko'pburchaklar jadvali

Ko'pburchak	Burchak ($\alpha$)	Tomoni ($a$)	Ichki radius ($r$)	Yuza ($S$)
Uchburchak ($n=3$)	$60^\circ$	$a = R\sqrt{3}$	$r = \frac{R}{2} = \frac{a\sqrt{3}}{6}$	$S = \frac{a^2\sqrt{3}}{4}$
Kvadrat ($n=4$)	$90^\circ$	$a = R\sqrt{2}$	$r = \frac{a}{2}$	$S = a^2 = 2R^2$
Oltiburchak ($n=6$)	$120^\circ$	$a = R$	$r = \frac{R\sqrt{3}}{2} = \frac{a\sqrt{3}}{2}$	$S = \frac{3a^2\sqrt{3}}{2}$

[!NOTE] Muntazam oltiburchak markazidan uchlariga kesmalar o'tkazilsa, u 6 ta teng tomonli uchburchakka ajraladi. Shuning uchun uning tomoni tashqi aylana radiusiga teng: $a = R$.

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Yechilgan misollar

1-misol. Muntazam o'nburchak burchaklari

Muntazam o'nburchakning ($n=10$) bitta ichki, tashqi va markaziy burchaklarini toping.

• Markaziy burchak: $\gamma = \frac{360^\circ}{10} = 36^\circ$.
• Tashqi burchak: $\beta = \frac{360^\circ}{10} = 36^\circ$.
• Ichki burchak: $\alpha = 180^\circ - \beta = 180^\circ - 36^\circ = 144^\circ$.

Javob: Markaziy $36^\circ$, tashqi $36^\circ$, ichki $144^\circ$.

2-misol. Tomonlar sonini aniqlash

Bitta ichki burchagi $135^\circ$ bo'lgan muntazam ko'pburchakning tomonlari soni va diagonallari sonini toping.

• Tashqi burchak: $\beta = 180^\circ - 135^\circ = 45^\circ$.
• Tomonlar soni: $n = \frac{360^\circ}{45^\circ} = 8$ (sakkizburchak).
• Diagonallar soni: $d = \frac{8 \cdot (8 - 3)}{2} = \frac{8 \cdot 5}{2} = 20$

Javob: Tomonlari 8 ta, diagonallari 20 ta.

3-misol. Muntazam oltiburchak elementlari

Radiusi $6\text{ sm}$ bo'lgan aylanaga ichki chizilgan muntazam oltiburchakning tomoni, apofemasi va yuzini hisoblang.

• Muntazam oltiburchakda tomon tashqi radiusga teng: $a = R = 6\text{ sm}$
• Apofema (ichki aylana radiusi): $r = \frac{a\sqrt{3}}{2} = \frac{6\sqrt{3}}{2} = 3\sqrt{3}\text{ sm}$
• Yuza: $S = \frac{3a^2\sqrt{3}}{2} = \frac{3 \cdot 6^2\sqrt{3}}{2} = \frac{3 \cdot 36\sqrt{3}}{2} = 54\sqrt{3}\text{ sm}^2$

Javob: Tomoni $6\text{ sm}$, apofemasi $3\sqrt{3}\text{ sm}$, yuzi $54\sqrt{3}\text{ sm}^2$.

4-misol. Kvadrat va aylanalar

Radiusi $4\text{ sm}$ bo'lgan aylanaga tashqi chizilgan kvadratning tomoni, perimetri va yuzini toping.

• Aylanaga tashqi chizilgan kvadrat uchun aylana radiusi ichki radius bo'ladi: $r = 4\text{ sm}$.
• Tomoni: $a = 2r = 2 \cdot 4 = 8\text{ sm}$.
• Perimetr: $P = 4a = 4 \cdot 8 = 32\text{ sm}$.
• Yuza: $S = a^2 = 8^2 = 64\text{ sm}^2$.

Javob: Tomoni $8\text{ sm}$, perimetri $32\text{ sm}$, yuzi $64\text{ sm}^2$.

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Mashq

1. Muntazam o'nikkiburchakning bitta ichki va bitta tashqi burchagini toping.
2. Ichki burchagi $140^\circ$ bo'lgan muntazam ko'pburchakning tomonlari sonini toping.
3. Muntazam oltiburchakning tomoni $10\text{ sm}$. Unga ichki va tashqi chizilgan aylanalar radiuslarini toping.
4. Tomoni $4\sqrt{3}\text{ sm}$ bo'lgan muntazam uchburchakning yuzini va unga ichki chizilgan aylana radiusini toping.
5. Kvadratning yuzi $36\text{ sm}^2$. Unga tashqi chizilgan aylananing radiusini hisoblang.
6. Muntazam oltiburchakka ichki chizilgan aylananing radiusi $6\text{ sm}$. Uning tomonini va tashqi aylana radiusini toping.
7. Muntazam oltiburchakning katta diagonali $16\text{ sm}$. Uning tomoni va kichik diagonalini toping.
8. Tomonlari soni 6 bo'lgan muntazam ko'pburchakning barcha diagonallari sonini hisoblang.
9. Tomoni $6\text{ sm}$ bo'lgan muntazam oltiburchakning yuzini hisoblang.
10. Muntazam ko'pburchakning bitta ichki burchagi tashqi burchagidan 5 marta katta. Ko'pburchakning tomonlari sonini toping.

Javoblar va yechim yo'llari

1. $\beta = 360^\circ : 12 = 30^\circ$, $\alpha = 180^\circ - 30^\circ = 150^\circ$. Javob: $150^\circ$ va $30^\circ$.
2. $\beta = 180^\circ - 140^\circ = 40^\circ$. $n = 360^\circ : 40^\circ = 9$. Javob: 9 ta.
3. $R = a = 10\text{ sm}$, $r = \frac{10\sqrt{3}}{2} = 5\sqrt{3}\text{ sm}$. Javob: $R = 10\text{ sm}$, $r = 5\sqrt{3}\text{ sm}$.
4. $S = \frac{(4\sqrt{3})^2\sqrt{3}}{4} = \frac{48\sqrt{3}}{4} = 12\sqrt{3}\text{ sm}^2$. $r = \frac{a\sqrt{3}}{6} = \frac{4\sqrt{3}\cdot\sqrt{3}}{6} = 2\text{ sm}$. Javob: $S = 12\sqrt{3}\text{ sm}^2$, $r = 2\text{ sm}$.
5. $a = \sqrt{36} = 6\text{ sm}$. $R = \frac{a\sqrt{2}}{2} = 3\sqrt{2}\text{ sm}$. Javob: $3\sqrt{2}\text{ sm}$.
6. $r = \frac{a\sqrt{3}}{2} \implies 6 = \frac{a\sqrt{3}}{2} \implies a = \frac{12}{\sqrt{3}} = 4\sqrt{3}\text{ sm}$. $R = a = 4\sqrt{3}\text{ sm}$. Javob: $a = 4\sqrt{3}\text{ sm}$, $R = 4\sqrt{3}\text{ sm}$.
7. Katta diagonal $d_1 = 2R = 2a = 16 \implies a = 8\text{ sm}$. Kichik diagonal $d_2 = a\sqrt{3} = 8\sqrt{3}\text{ sm}$. Javob: Tomoni $8\text{ sm}$, kichik diagonal $8\sqrt{3}\text{ sm}$.
8. $d = \frac{6 \cdot (6 - 3)}{2} = \frac{6 \cdot 3}{2} = 9$. Javob: 9 ta.
9. $S = \frac{3 \cdot 6^2\sqrt{3}}{2} = 54\sqrt{3}\text{ sm}^2$. Javob: $54\sqrt{3}\text{ sm}^2$.
10. $\alpha = 5\beta$. $\alpha + \beta = 180^\circ \implies 6\beta = 180^\circ \implies \beta = 30^\circ$. $n = 360^\circ : 30^\circ = 12$. Javob: 12 ta.

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Keng tarqalgan xatolar

1. Barcha ko'pburchaklarda $a = R$ deb hisoblash. ✗ Uchburchak yoki kvadratda ham tomonni tashqi radiusga teng deb olish. ✓ Tomoni tashqi radiusga teng bo'lishi ($a = R$) faqat va faqat muntazam oltiburchak uchungina xosdir.

2. Ichki burchak yig'indisi formulasida $(n-2)$ ni unutish. ✗ Yig'indini $n \cdot 180^\circ$ deb yozish. ✓ Qavariq ko'pburchakning ichki burchaklari yig'indisi doimo $(n - 2) \cdot 180^\circ$ ga teng.

3. Tashqi burchaklar yig'indisini $n$ ga ko'paytirish. ✗ Tomonlar soni ko'paygani sari tashqi burchaklar yig'indisi ham ortadi deb o'ylash. ✓ Har qanday qavariq ko'pburchakda tashqi burchaklar yig'indisi doimo o'zgarmas $360^\circ$ dir.

4. Ichki va tashqi radiuslarni adashtirish. ✗ Muntazam ko'pburchakda $r > R$ deb hisob-kitob qilish. ✓ Doimo tashqi radius ichki radiusdan kattadir: $R > r$ ($R$ burchak uchiga boradi, $r$ tomon o'rtasiga tushadi).

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DTM uslubidagi test

Ushbu mavzuni mustahkamlash uchun berilgan 5 ta test savolini ishlab, javoblaringizni tekshiring.

Keyingi qadam

Aylana va ko'pburchaklar mavzularini to'liq yakunladik. Keyingi bo'limda tekislikdagi yo'naltirilgan kesmalar — vektorlar va koordinatalar metodi bilan tanishamiz.