Find the perimeter $ P $ of a hexagon if area $A = 486 \sqrt{ 3 }$.

$P = 108$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ A = 486 \sqrt{ 3 } $ we have:

$$ 486 \sqrt{ 3 } = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 486 \sqrt{ 3 } \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 972 \sqrt{ 3 } = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 972 \sqrt{ 3 } }{ 3 \sqrt{ 3 } } $$ $$ a^2 = 324 $$ $$ a = \sqrt{ 324 } $$ $$ a = 18 $$

STEP 2: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 6 \cdot a $$

After substituting $ a = 18 $ we have:

$$ P = 6 \cdot 18 $$ $$ P = 108 $$

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