Find the length $ l $ of a triangular prism if height $h = 11$ and volume $V = 115.5$.

$l = 0.8267$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ h = 11 $ we have:

$$ 11 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 11 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 22 $$ $$ a = \dfrac{ 22 }{ \sqrt{ 3 } } $$ $$ a = \frac{ 22 \sqrt{ 3}}{ 3 } $$

STEP 2: find length $ l $

To find length $ l $ use formula:

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

After substituting $ V = 115.5 $ and $ a = \frac{ 22 \sqrt{ 3}}{ 3 } $ we have:

$$ 115.5 = \dfrac{ \sqrt{ 3 } \cdot \frac{ 22 \sqrt{ 3}}{ 3 } ^{ 2 } \cdot l }{ 2 } $$$$ 115.5 \cdot 2 = \sqrt{ 3 } \cdot \frac{ 22 \sqrt{ 3}}{ 3 } ^{ 2 } \cdot l $$$$ 231 = \sqrt{ 3 } \cdot \frac{ 22 \sqrt{ 3}}{ 3 } ^{ 2 } \cdot l $$$$ 231 = \frac{ 484 \sqrt{ 3}}{ 3 } \cdot l $$$$ l = \dfrac{ 231 }{ \frac{ 484 \sqrt{ 3}}{ 3 } } $$$$ l = 0.8267 $$

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