Find the area $ A $ of a hexagon if long diagonal $d = \frac{ 12 }{ 5 }$.

$A = \frac{ 54 \sqrt{ 3}}{ 25 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ d = 2 \cdot a $$

After substituting $ d = \frac{ 12 }{ 5 } $ we have:

$$ \frac{ 12 }{ 5 } = 2 \cdot a $$ $$ a = \dfrac{ \frac{ 12 }{ 5 } }{ 2 } $$ $$ a = \frac{ 6 }{ 5 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = \frac{ 6 }{ 5 } $ we have:

$$ A = \dfrac{ 3 \sqrt{ 3 } \cdot \left(\frac{ 6 }{ 5 }\right)^2 }{ 2 } $$ $$ A = \dfrac{ 3 \sqrt{ 3 } \cdot \frac{ 36 }{ 25 } }{ 2 }$$ $$ A = \dfrac{ \frac{ 108 \sqrt{ 3}}{ 25 } }{ 2 }$$ $$ A = \frac{ 54 \sqrt{ 3}}{ 25 } $$

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