Find the short diagonal $ s $ of a hexagon if area $A = 70$.

$s = 8.9905$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ A = 70 $ we have:

$$ 70 = \dfrac{ 3 \sqrt{ 3 } \cdot a^2 }{ 2 }$$ $$ 70 \cdot 2 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ 140 = 3 \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 140 }{ 3 \sqrt{ 3 } } $$ $$ a^2 = \frac{ 140 \sqrt{ 3}}{ 9 } $$ $$ a = \sqrt{ \frac{ 140 \sqrt{ 3}}{ 9 } } $$$$ a \approx 5.1907 $$

STEP 2: find diagonal $ s $

To find diagonal $ s $ use formula:

$$ s = \sqrt{ 3 } \cdot a $$

After substituting $ a = 5.1907 $ we have:

$$ s = \sqrt{ 3 } \cdot 5.1907 $$ $$ s \approx 8.9905 $$

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