STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $A = 12\, \text{cm}$ we have:

$$ 12\, \text{cm} = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 12\, \text{cm} \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 24\, \text{cm} = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 24\, \text{cm} }{ 3 \sqrt{ 3 } } $$ $$ a^2 = \frac{ 8 \sqrt{ 3}}{ 3 }\, \text{cm} $$ $$ a = \sqrt{ \frac{ 8 \sqrt{ 3}}{ 3 }\, \text{cm} } $$$$ a \approx 2.1491 $$

STEP 2:

The radius of the circumcircle is equal the side of the hexagon. $$ R = 2.1491 $$

Hexagon Calculator