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 result:

S =  48.808

#### Solution:

$n=15 \ \\ r=4 \ \\ \ \\ α=\dfrac{ 2 \pi }{ 2 \cdot \ n }=\dfrac{ 2 \cdot \ 3.1416 }{ 2 \cdot \ 15 } \doteq 0.2094 \ \\ \ \\ \sin α=a/2 : r \ \\ \ \\ a=2 \cdot \ r \cdot \ \sin(α)=2 \cdot \ 4 \cdot \ \sin(0.2094) \doteq 1.6633 \ \\ \cos α=h:r \ \\ \ \\ h=r \cdot \ \cos(α)=4 \cdot \ \cos(0.2094) \doteq 3.9126 \ \\ \ \\ S_{1}=\dfrac{ a \cdot \ h }{ 2 }=\dfrac{ 1.6633 \cdot \ 3.9126 }{ 2 } \doteq 3.2539 \ \\ \ \\ S=n \cdot \ S_{1}=15 \cdot \ 3.2539=48.808$

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