Pentagon

Calculate the length of side, circumference and area of a regular pentagon, which is inscribed in a circle with radius r = 6 cm.

Correct answer:

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

Step-by-step explanation:

$$\begin{gathered} r=6 \\ a=2\cdot r\cdot \sin (\pi \mathrm{/}5)=2\cdot 6\cdot \sin (3.1416\mathrm{/}5)=7.0534\text{ cm} \end{gathered}$$

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

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

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