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Vesna [10]
3 years ago
12

The area of a given hexagon is equal to the area of an equilateral triangle whose perimeter is 36 inches. Find the length of a s

ide of the regular hexagon.The area of a given hexagon is equal to the area of an equilateral triangle whose perimeter is 36 inches. Find the length of a side of the regular hexagon.The area of a given hexagon is equal to the area of an equilateral triangle whose perimeter is 36 inches. Find the length of a side of the regular hexagon. Please answer in square root form.
Mathematics
2 answers:
garri49 [273]3 years ago
5 0
\bf \textit{area of an equilateral triangle}\\\\
A=\cfrac{s^2\sqrt{3}}{4}\qquad 
\begin{cases}
s=\textit{length of a side}\\
----------\\
perimeter=s+s+s\\
perimiter=3s\\
36=3s\\
\frac{36}{3}=s\\
12=s
\end{cases}
\\\\\\
A=\cfrac{12^2\sqrt{3}}{4}\implies A=36\sqrt{3}

ok.. .based on a side of 12, that's the area of the equilateral triangle, now, the hexagon has the same area... so... let's use the area of a polygon to see what's the length of a side

\bf \textit{area of a regular polygon}\\\\
A=\cfrac{1}{4}ns^2\ cot\left( \frac{180}{n} \right)\qquad 
\begin{cases}
n=\textit{number of sides}\\
s=\textit{length of one side}\\
----------\\
n=6\\
A=36\sqrt{3}
\end{cases}
\\\\\\
36\sqrt{3}=\cfrac{1}{4}\cdot 6\cdot s^2\cdot cot\left( \frac{180}{6} \right)\implies 36\sqrt{3}=\cfrac{1}{4}\cdot 6\cdot s^2\cdot \sqrt{3}
\\\\\\
36\sqrt{3}=\cfrac{6s^2\sqrt{3}}{4}\implies \cfrac{4\cdot 36\sqrt{3}}{6\sqrt{3}}=s^2
\\\\\\
24=s^2\implies \sqrt{24}=s\implies 2\sqrt{6}=s


now, in case you want to check how much is the cot(30°), check your Unit Circle, recall, cotangent is cosine/sine
KonstantinChe [14]3 years ago
3 0

Answer:

2squrt6 for short

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