Decagon

Calculate the area and circumference of the regular decagon when its radius of a circle circumscribing is R = 1m

Correct result:

S =  2.939 m2
o =  6.18 m

Solution:

$R=1 \ \text{m} \ \\ n=10 \ \\ \ \\ A=360 / n=360 / 10=36 \ ^\circ \ \\ \ \\ \sin A/2=\dfrac{ a/2 }{ R } \ \\ \ \\ a=2 \cdot \ R \cdot \ \sin(A/2)=2 \cdot \ 1 \cdot \ \sin(36/2) \doteq 0.618 \ \text{m} \ \\ \ \\ r=\sqrt{ R^2 - (a/2)^2 }=\sqrt{ 1^2 - (0.618/2)^2 } \doteq 0.9511 \ \text{m} \ \\ \ \\ S_{1}=\dfrac{ r \cdot \ a }{ 2 }=\dfrac{ 0.9511 \cdot \ 0.618 }{ 2 } \doteq 0.2939 \ \text{m}^2 \ \\ S=n \cdot \ S_{1}=10 \cdot \ 0.2939=2.939 \ \text{m}^2$
$o=n \cdot \ a=10 \cdot \ 0.618=6.18 \ \text{m}$

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