Decadal - flower bed

The castle park includes a flower bed in the shape of a regular decagon with an area of 432.8 m². Determine the distance between adjacent vertices of the flower bed.

Final Answer:

a =  7.5 m

Step-by-step explanation:

$$\begin{gathered} n=10 \\ S=432.8{\text{m}}^2 \\ \alpha =360/n=360/10=36{}^{\circ } \\ \beta =\alpha /2=36/2=18{}^{\circ } \\ {S}_1=S/n=432.8/10=\frac{1082}{25}=43.28{\text{m}}^2 \\ {S}_2=S_1/2=43.28/2=\frac{541}{25}=21.64{\text{m}}^2 \\ \tan \beta =y:x \\ {S}_2=\frac{y\cdot x}{2} \\ {S}_2=\frac{y\cdot y/\tan \beta }{2} \\ y=\sqrt{2\cdot S_2\cdot \tan \beta }=\sqrt{2\cdot S_2\cdot \tan 18°}=\sqrt{2\cdot 21.64\cdot \tan 18°}=\sqrt{2\cdot 21.64\cdot 0.32492}=3.75\text{m} \\ a=2\cdot y=2\cdot 3.75=7.5\text{m} \end{gathered}$$

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