Find the perimeter $ P $ of an equilateral triangle if area $A = 27 \sqrt{ 3 }$.

$P = 18 \sqrt{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ A = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 }$$

After substituting $ A = 27 \sqrt{ 3 } $ we have:

$$ 27 \sqrt{ 3 } = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 }$$ $$ 27 \sqrt{ 3 } \cdot 4 = \sqrt{ 3 } \cdot a^2 $$ $$ 108 \sqrt{ 3 } = \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 108 \sqrt{ 3 } }{ \sqrt{ 3 } } $$ $$ a^2 = 108 $$ $$ a = \sqrt{ 108 } $$ $$ a = 6 \sqrt{ 3 } $$

STEP 2: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 3 \cdot a $$

After substituting $ a = 6 \sqrt{ 3 } $ we have:

$$ P = 3 \cdot 6 \sqrt{ 3 } $$ $$ P = 18 \sqrt{ 3 } $$

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