Pentadecagon

Calculate the content of a regular 15-sides polygon inscribed in a circle with radius r = 4. Express the result to two decimal places.

Correct answer:

S =  48.8084

Step-by-step explanation:

$$\begin{gathered} n=15 \\ r=4 \\ \alpha ={\displaystyle \frac{2\pi }{2\cdot n}}={\displaystyle \frac{2\cdot 3.1416}{2\cdot 15}}\doteq 0.2094 \\ \sin \alpha =a\mathrm{/}2:r \\ a=2\cdot r\cdot \sin (\alpha )=2\cdot 4\cdot \sin (0.2094)\doteq 1.6633 \\ \cos \alpha =h:r \\ h=r\cdot \cos (\alpha )=4\cdot \cos (0.2094)\doteq 3.9126 \\ {S}_1={\displaystyle \frac{a\cdot h}{2}}={\displaystyle \frac{1.6633\cdot 3.9126}{2}}\doteq 3.2539 \\ S=n\cdot S_1=15\cdot 3.2539=48.8084 \end{gathered}$$

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