Five-gon

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

Correct answer:

a =  7.0534 cm
o =  35.2671 cm
S =  85.5951 cm²

Step-by-step explanation:

$$\begin{gathered} n=5 \\ R=6\text{ cm } \\ \mathrm{\Phi }=360\mathrm{/}(2\cdot n)=360\mathrm{/}(2\cdot 5)=36\text{ ° } \\ \sin \mathrm{\Phi }=a\mathrm{/}2:R \\ a=2\cdot R\cdot \sin \mathrm{\Phi }\mathrm{°}=2\cdot R\cdot \sin 36\mathrm{°}=2\cdot 6\cdot \sin 36\mathrm{°}=2\cdot 6\cdot 0.587785=7.053=7.0534\text{ cm} \end{gathered}$$

$o=n\cdot a=5\cdot 7.0534=35.2671\text{ cm}$

$$\begin{gathered} h=\sqrt{R^2-(a\mathrm{/}2{)}^2}=\sqrt{6^2-(7.0534\mathrm{/}2{)}^2}\doteq 4.8541\text{ cm } \\ {S}_1={\displaystyle \frac{a\cdot h}{2}}={\displaystyle \frac{7.0534\cdot 4.8541}{2}}\doteq 17.119{\text{cm}}^2 \\ S=n\cdot S_1=5\cdot 17.119=85.5951{\text{cm}}^2 \end{gathered}$$

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