Find the perimeter $ P $ of a hexagon if area $A = 1$.

$P = 3.7224$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ A = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$

After substituting $ A = 1 $ we have:

$$ 1 = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 1 \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 2 }{ 3 \sqrt{ 3 } } $$ $$ a^2 = \frac{ 2 \sqrt{ 3}}{ 9 } $$ $$ a = \sqrt{ \frac{ 2 \sqrt{ 3}}{ 9 } } $$$$ a \approx 0.6204 $$

STEP 2: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 6 \cdot a $$

After substituting $ a = 0.6204 $ we have:

$$ P = 6 \cdot 0.6204 $$ $$ P \approx 3.7224 $$

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