Find the incircle radius $ r $ of a hexagon if area $A = 124.7077$.

$r = 6$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ A = 124.7077 $ we have:

$$ 124.7077 = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 124.7077 \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 249.4153 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 249.4153 }{ 3 \sqrt{ 3 } } $$ $$ a^2 = 48 $$ $$ a = \sqrt{ 48 } $$$$ a \approx 6.9282 $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

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

After substituting $ a = 6.9282 $ we have:

$$ r = \dfrac{ \sqrt{ 3 } \cdot 6.9282 }{ 2 } $$$$ r = \dfrac{ 12 }{ 2 } $$ $$ r \approx 6 $$

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