Ko'pburchak va aylana

Ushbu darsda ko‘pburchak va aylana orasidagi bog‘lanish haqida so‘z yuritamiz. Ayniqsa muntazam ko‘pburchak — barcha tomonlari va burchaklari teng bo‘lgan ko‘pburchak — bilan aylana o‘rtasidagi go‘zal aloqani o‘rganamiz. Har bir muntazam ko‘pburchak uchun ham ichki, ham tashqi aylana mavjud.

§1. Muntazam ko‘pburchak, ichki va tashqi aylana

Ta’rif. Barcha tomonlari o‘zaro teng va barcha burchaklari o‘zaro teng bo‘lgan ko‘pburchak muntazam ko‘pburchak deyiladi.

Har qanday muntazam $n$-burchakning bitta markazi bo‘lib, undan barcha uchlargacha masofa bir xil ($R$ — tashqi radius), barcha tomonlargacha masofa ham bir xil ($r$ — ichki radius yoki apofema).

Qoida. Muntazam $n$-burchakning markaziy burchagi: $$ \varphi = \frac{360^\circ}{n}, $$ ichki burchagi esa $$ \alpha = \frac{(n-2)\cdot 180^\circ}{n}. $$

§2. Tomon, apofema va radius bog‘lanishi

Markazni ikki qo‘shni uchga tutashtirsak, teng yonli uchburchak hosil bo‘ladi, uning uchidagi burchagi $\varphi=\tfrac{360^\circ}{n}$. Shu uchburchakdan tomon va apofemani topamiz.

Qoida. Muntazam $n$-burchak uchun: $$ a = 2R\sin\frac{180^\circ}{n}, \qquad r = R\cos\frac{180^\circ}{n}, $$ bu yerda $a$ — tomon, $r$ — apofema (ichki radius).

Qoida. Yuzasi: $$ S = \frac{1}{2},P\cdot r = \frac{1}{2},n,a,r, $$ bu yerda $P=na$ — perimetr.

Eslatma. Bu formulalar $n$ katta bo‘lganda ko‘pburchakning aylanaga tobora yaqinlashishini ko‘rsatadi: $n\to\infty$ da perimetr $2\pi R$ ga, yuza esa $\pi R^2$ ga intiladi.

Misol. Muntazam oltiburchakning (heksagon) tashqi radiusi $R=6$. Tomonini toping.

Yechish. $n=6$ uchun:

Ko‘ramizki, muntazam oltiburchakda tomon radiusga teng.

Javob. $a = 6$.

Misol. Muntazam oltiburchakning tomoni $a=4$. Uning yuzasini toping.

Yechish. Oltiburchakda $R=a=4$. Apofema:

Perimetr $P = 6\cdot4 = 24$. Yuza:

Javob. $S = 24\sqrt3$.

Misol. Muntazam kvadratning (to‘rtburchak) tashqi radiusi $R=5\sqrt2$. Tomonini va ichki radiusini toping.

Yechish. $n=4$ uchun:

Javob. $a = 10$, $r = 5$.

§3. Ichki burchakni hisoblash

Misol. Muntazam beshburchak (pentagon) ichki burchagini toping.

Yechish. $n=5$ uchun:

Javob. $108^\circ$.

§ Lug‘at

muntazam ko‘pburchak — regular polygon
markaziy burchak — central angle
ichki burchak — interior angle
apofema — apothem
tashqi radius — circumradius
perimetr — perimeter