STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $A = 72 \sqrt{ 3 }\, \text{cm}$ we have:

$$ 72 \sqrt{ 3 }\, \text{cm} = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 72 \sqrt{ 3 }\, \text{cm} \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 144 \sqrt{ 3 }\, \text{cm} = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 144 \sqrt{ 3 }\, \text{cm} }{ 3 \sqrt{ 3 } } $$ $$ a^2 = 48\, \text{cm} $$ $$ a = \sqrt{ 48\, \text{cm} } $$ $$ a = 4 \sqrt{ 3 } $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

$$ r = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$

After substituting $a = 4 \sqrt{ 3 }\, \text{cm}^0$ we have:

$$ r = \dfrac{ \sqrt{ 3 } \cdot 4 \sqrt{ 3 } }{ 2 } $$$$ r = \dfrac{ 12 }{ 2 } $$ $$ r = 6 $$

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