Find the area $ A $ of a hexagon if incircle radius $r = \frac{ 52 }{ 5 }$.

$A = \frac{ 5408 \sqrt{ 3}}{ 25 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ r = \frac{ 52 }{ 5 } $ we have:

$$ \frac{ 52 }{ 5 } = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = \frac{ 52 }{ 5 } \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = \frac{ 104 }{ 5 } $$ $$ a = \dfrac{ \frac{ 104 }{ 5 } }{ \sqrt{ 3 } } $$ $$ a = \frac{ 104 \sqrt{ 3}}{ 15 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = \frac{ 104 \sqrt{ 3}}{ 15 } $ we have:

$$ A = \dfrac{ 3 \sqrt{ 3 } \cdot \left( \frac{ 104 \sqrt{ 3}}{ 15 } \right)^2 }{ 2 } $$ $$ A = \dfrac{ 3 \sqrt{ 3 } \cdot \frac{ 10816 }{ 75 } }{ 2 }$$ $$ A = \dfrac{ \frac{ 10816 \sqrt{ 3}}{ 25 } }{ 2 }$$ $$ A = \frac{ 5408 \sqrt{ 3}}{ 25 } $$

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