# 30-gon

At a regular 30-gon the radius of the inscribed circle is 15cm. Find the "a" side size, circle radius "R", circumference, and content area.

Result

a =  3.153 cm
R =  15.083 cm
o =  94.59 cm
S =  709.425 cm2

#### Solution:

$r=15 \ \\ n=30 \ \\ A=360/n=360/30=12 \ \\ \cos A/2=r/R \ \\ R=r / \cos( A/2 )=15 / \cos( 12/2 ) \doteq 15.0826 \ \\ (a/2)^2=R^2 - r^2 \ \\ x=\sqrt{ R^2-r^2 }=\sqrt{ 15.0826^2-15^2 } \doteq 1.5766 \ \\ a=2 \cdot \ x=2 \cdot \ 1.5766 \doteq 3.1531 \doteq 3.153 \ \text{cm}$
$o=n \cdot \ a=30 \cdot \ 3.1531=\dfrac{ 9459 }{ 100 }=94.59 \ \text{cm}$
$S_{1}=a \cdot \ r / 2=3.1531 \cdot \ 15 / 2=\dfrac{ 9459 }{ 400 }=23.6475 \ \\ S=n \cdot \ S_{1}=30 \cdot \ 23.6475=\dfrac{ 28377 }{ 40 }=709.425 \ \text{cm}^2$

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