Ko'pburchaklar va muntazam ko'pburchaklar: matematika darsi

Muntazam ko'pburchakning ichki burchagi $144^\circ$ bo'lsa, tomonlari soni $10$ ta, diagonallari esa $35$ ta. Buni bir qadamda topish mumkin, agar ichki burchak bilan emas, tashqi burchak bilan ishlansa. Bu darsda burchaklar, diagonallar, muntazam ko'pburchak formulalari, ayniqsa muntazam oltiburchak va sakkizburchak beriladi.

Ko'pburchak va burchaklar yig'indisi

Qavariq ko'pburchak deb har bir tomoni yotgan to'g'ri chiziqning bir tomonida joylashgan ko'pburchakka aytiladi. Uning har bir ichki burchagi $180^\circ$ dan kichik.

Bir uchdan chiqqan diagonallar oltiburchakni

4

ta uchburchakka ajratadi

\Sigma=(n-2)\cdot 180^\circ

Bir uchdan

n-3

ta diagonal chiqadi va ular

n-2

ta uchburchak hosil qiladi

\Sigma_{\text{tashqi}}=360^\circ

Har bir uchda bittadan tashqi burchak olinganda,

n

ga bog'liq emas. Har bir uchda ichki va tashqi burchak yig'indisi

180^\circ

N=\frac{n(n-3)}{2}

Har bir uchdan

n-3

ta diagonal, har bir diagonal ikki marta sanaladi

Qavariq ko'pburchakda o'tkir burchaklar ko'pi bilan $3$ ta bo'ladi, chunki o'tkir burchakning tashqi burchagi $90^\circ$ dan katta, tashqi burchaklar yig'indisi esa $360^\circ$ . Burchaklar progressiya yoki nisbat bilan berilsa, yig'indi tenglamasi tuziladi. Keyin eng katta burchak $180^\circ$ dan kichikligi tekshiriladi.

Muntazam ko'pburchak

Muntazam ko'pburchak deb barcha tomonlari teng va barcha burchaklari teng bo'lgan qavariq ko'pburchakka aytiladi.

Ikkala shart ham kerak: rombda tomonlar teng, to'g'ri to'rtburchakda burchaklar teng, lekin ular muntazam emas. Faqat uchburchakda bitta shart yetarli.

\beta=\frac{360^\circ}{n}

\alpha=180^\circ-\beta

\alpha=\frac{(n-2)\cdot 180^\circ}{n}

\beta

— tashqi burchak, u markaziy burchakka teng;

\alpha

— ichki burchak

Har qanday burchak muntazam ko'pburchakning burchagi bo'lavermaydi: $\dfrac{360^\circ}{\beta}$ butun va kamida $3$ bo'lishi kerak. Masalan, $\alpha=130^\circ$ bo'lishi mumkin emas, chunki $\beta=50^\circ$ va $360$ soni $50$ ga bo'linmaydi.

$n$	$\Sigma$	$\alpha$	$\beta$	$N$
$n$ $3$	$\Sigma$ $180^\circ$	$\alpha$ $60^\circ$	$\beta$ $120^\circ$	$N$ $0$
$n$ $4$	$\Sigma$ $360^\circ$	$\alpha$ $90^\circ$	$\beta$ $90^\circ$	$N$ $2$
$n$ $5$	$\Sigma$ $540^\circ$	$\alpha$ $108^\circ$	$\beta$ $72^\circ$	$N$ $5$
$n$ $6$	$\Sigma$ $720^\circ$	$\alpha$ $120^\circ$	$\beta$ $60^\circ$	$N$ $9$
$n$ $8$	$\Sigma$ $1080^\circ$	$\alpha$ $135^\circ$	$\beta$ $45^\circ$	$N$ $20$
$n$ $9$	$\Sigma$ $1260^\circ$	$\alpha$ $140^\circ$	$\beta$ $40^\circ$	$N$ $27$
$n$ $10$	$\Sigma$ $1440^\circ$	$\alpha$ $144^\circ$	$\beta$ $36^\circ$	$N$ $35$
$n$ $12$	$\Sigma$ $1800^\circ$	$\alpha$ $150^\circ$	$\beta$ $30^\circ$	$N$ $54$

Ichki va tashqi burchak munosabati berilsa, $\alpha+\beta=180^\circ$ bilan sistema tuziladi. $\alpha=5\beta$ bo'lsa, $\beta=30^\circ$ , $n=12$ . $\alpha-\beta=90^\circ$ bo'lsa, $\beta=45^\circ$ , $n=8$ .

Radiuslar, tomon va yuz

OMA

to'g'ri burchakli uchburchak:

R

gipotenuza,

\angle AOM=\dfrac{180^\circ}{n}

a=2R\sin\frac{180^\circ}{n}

r=R\cos\frac{180^\circ}{n}

R^2=r^2+\frac{a^2}{4}

R

— tashqi chizilgan aylana radiusi,

r

— ichki chizilgan aylana radiusi

S=\frac{1}{2}Pr=\frac{1}{2}nar

S=\frac{1}{2}nR^2\sin\frac{360^\circ}{n}

Markazdan uchlarga kesmalar ko'pburchakni

n

ta teng uchburchakka ajratadi

$n$	$R$	$r$	$S$
$n$ $3$	$R$ $\dfrac{a\sqrt{3}}{3}$	$r$ $\dfrac{a\sqrt{3}}{6}$	$S$ $\dfrac{a^2\sqrt{3}}{4}$
$n$ $4$	$R$ $\dfrac{a\sqrt{2}}{2}$	$r$ $\dfrac{a}{2}$	$S$ $a^2$
$n$ $6$	$R$ $a$	$r$ $\dfrac{a\sqrt{3}}{2}$	$S$ $\dfrac{3a^2\sqrt{3}}{2}$
$n$ $8$	$R$ $R^2=\dfrac{a^2}{2-\sqrt{2}}$	$r$ $r=\dfrac{a(1+\sqrt{2})}{2}$	$S$ $2(1+\sqrt{2})a^2=2\sqrt{2}R^2$
$n$ $12$	$R$ $a=2R\sin 15^\circ$	$r$ $r=R\cos 15^\circ$	$S$ $3R^2$

Tengdosh shakllar yuzlari teng shakllardir. Masalan, oltiburchak tomoni $2\sqrt{6}$ bo'lsa, $S=\dfrac{3\cdot 24\sqrt{3}}{2}=36\sqrt{3}$ , unga tengdosh muntazam uchburchak uchun $\dfrac{x^2\sqrt{3}}{4}=36\sqrt{3}$ , $x=12$ .

Muntazam oltiburchak

Oltita muntazam uchburchak, har birining tomoni

a

Tomoni $a$ bo'lgan oltiburchakda	Qiymat
Katta diagonal $AD$ , $BE$ , $CF$	$2a$
Kichik diagonal $AC$ , $BD$ , $CE$ va boshqalar	$a\sqrt{3}$
Qarama-qarshi tomonlar orasidagi masofa	$a\sqrt{3}$
$C$ dan $AE$ gacha masofa	$\dfrac{3a}{2}$
Bitta markaziy uchburchak yuzi	$\dfrac{S}{6}$
$ABC$ uchburchak yuzi	$\dfrac{S}{6}$
$ACE$ muntazam uchburchak yuzi	$\dfrac{S}{2}$
$ABCD$ trapetsiya yuzi (katta diagonal ajratgan)	$\dfrac{S}{2}$
$ABDE$ to'g'ri to'rtburchak yuzi	$\dfrac{2S}{3}$

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