# Five-gon

Calculate the side a, the circumference and the area of the regular 5-angle if Rop = 6cm.

Result

a =  7.053 cm
o =  35.265 cm
S =  85.593 cm2

#### Solution:

$n=5 \ \\ R=6 \ \text{cm} \ \\ \ \\ Φ=360/(2 \cdot \ n)=360/(2 \cdot \ 5)=36 \ ^\circ \ \\ \ \\ \sin Φ=a/2 : R \ \\ \ \\ a=2 \cdot \ R \cdot \ \sin(Φ)=2 \cdot \ 6 \cdot \ \sin(36^\circ ) \doteq 7.0534 \doteq 7.053 \ \text{cm}$
$o=n \cdot \ a=5 \cdot \ 7.0534=\dfrac{ 7053 }{ 200 }=35.265 \ \text{cm}$
$h=\sqrt{ R^2-(a/2)^2 }=\sqrt{ 6^2-(7.0534/2)^2 } \doteq 4.8543 \ \text{cm} \ \\ \ \\ S_{1}=\dfrac{ a \cdot \ h }{ 2 }=\dfrac{ 7.0534 \cdot \ 4.8543 }{ 2 } \doteq 17.1185 \ \text{cm}^2 \ \\ \ \\ S=n \cdot \ S_{1}=5 \cdot \ 17.1185 \doteq 85.5927 \doteq 85.593 \ \text{cm}^2$

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