Find the area $ A $ of a hexagon if incircle radius $r = 12$.

$A = 288 \sqrt{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ r = 12 $ we have:

$$ 12 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 12 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 24 $$ $$ a = \dfrac{ 24 }{ \sqrt{ 3 } } $$ $$ a = 8 \sqrt{ 3 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 8 \sqrt{ 3 } $ we have:

$$ A = \dfrac{ 3 \sqrt{ 3 } \cdot \left(8 \sqrt{ 3 }\right)^2 }{ 2 } $$ $$ A = \dfrac{ 3 \sqrt{ 3 } \cdot 192 }{ 2 }$$ $$ A = \dfrac{ 576 \sqrt{ 3 } }{ 2 }$$ $$ A = 288 \sqrt{ 3 } $$

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