Find the height $ h $ of a triangular prism if length $l = 3$ and volume $V = 4.5$.

$h = 1.1398$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ V = 4.5 $ and $ l = 3 $ we have:

$$ 4.5 = \dfrac{ \sqrt{ 3 } \cdot a ^{ 2 } \cdot 3 }{ 2 } $$$$ 4.5 \cdot 2 = \sqrt{ 3 } \cdot a ^{ 2 } \cdot 3 $$$$ 9 = \sqrt{ 3 } \cdot a ^{ 2 } \cdot 3 $$$$ 9 = 3 \sqrt{ 3 } \cdot a ^{ 2 } $$$$ a ^{ 2 } = \dfrac{ 9}{ 3 \sqrt{ 3 } } $$$$ a ^{ 2 } = 1.7321 $$$$ a = \sqrt{ 1.7321 } $$$$ a = 1.3161 $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ a = 1.3161 $ we have:

$$ h = \dfrac{ \sqrt{ 3 } \cdot 1.3161 }{ 2 } $$$$ h = \dfrac{ 2.2795 }{ 2 } $$ $$ h \approx 1.1398 $$

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