Find the incircle radius $ r $ of a hexagon if area $A = \frac{ 15177 }{ 50 }$.

$r = 9.3608$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ A = \frac{ 15177 }{ 50 } $ we have:

$$ \frac{ 15177 }{ 50 } = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ \frac{ 15177 }{ 50 } \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ \frac{ 15177 }{ 25 } = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ \frac{ 15177 }{ 25 } }{ 3 \sqrt{ 3 } } $$ $$ a^2 = \frac{ 5059 \sqrt{ 3}}{ 75 } $$ $$ a = \sqrt{ \frac{ 5059 \sqrt{ 3}}{ 75 } } $$$$ a \approx 10.8089 $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

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

After substituting $ a = 10.8089 $ we have:

$$ r = \dfrac{ \sqrt{ 3 } \cdot 10.8089 }{ 2 } $$$$ r = \dfrac{ 18.7216 }{ 2 } $$ $$ r \approx 9.3608 $$

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