Find the incircle radius $ r $ of a hexagon if area $A = \frac{ 13479 }{ 25 }$.

$r = 12.4757$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ A = \frac{ 13479 }{ 25 } $ we have:

$$ \frac{ 13479 }{ 25 } = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ \frac{ 13479 }{ 25 } \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ \frac{ 26958 }{ 25 } = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ \frac{ 26958 }{ 25 } }{ 3 \sqrt{ 3 } } $$ $$ a^2 = \frac{ 8986 \sqrt{ 3}}{ 75 } $$ $$ a = \sqrt{ \frac{ 8986 \sqrt{ 3}}{ 75 } } $$$$ a \approx 14.4057 $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

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

After substituting $ a = 14.4057 $ we have:

$$ r = \dfrac{ \sqrt{ 3 } \cdot 14.4057 }{ 2 } $$$$ r = \dfrac{ 24.9513 }{ 2 } $$ $$ r \approx 12.4757 $$

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