STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $A = 100\, \text{cm}$ we have:

$$ 100\, \text{cm} = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 100\, \text{cm} \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 200\, \text{cm} = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 200\, \text{cm} }{ 3 \sqrt{ 3 } } $$ $$ a^2 = \frac{ 200 \sqrt{ 3}}{ 9 }\, \text{cm} $$ $$ a = \sqrt{ \frac{ 200 \sqrt{ 3}}{ 9 }\, \text{cm} } $$$$ a \approx 6.204 $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

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

After substituting $a = 6.204\, \text{cm}^0$ we have:

$$ r = \dfrac{ \sqrt{ 3 } \cdot 6.204 }{ 2 } $$$$ r = \dfrac{ 10.7457 }{ 2 } $$ $$ r \approx 5.3728 $$

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