Find the length $ l $ of a triangular prism if height $h = 3$ and area $A = 30$.

$l = 1.8868$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ h = 3 $ we have:

$$ 3 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 3 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 6 $$ $$ a = \dfrac{ 6 }{ \sqrt{ 3 } } $$ $$ a = 2 \sqrt{ 3 } $$

STEP 2: find length $ l $

To find length $ l $ use formula:

$$ A = \frac{ a^2 \sqrt{3}}{2} + 3 a l $$

After substituting $ A = 30 $ and $ a = 2 \sqrt{ 3 } $ we have:

$$ 30 = \frac{ 2 \sqrt{ 3 }^2 \sqrt{3}}{2} + 3 \cdot 2 \sqrt{ 3 } \cdot l $$$$ 30 = \frac{ 12 \sqrt{3}}{2} + 6 \sqrt{ 3 } \cdot l $$$$ 2 \cdot 30 = 12 \sqrt{ 3 } + 2 \cdot 6 \sqrt{ 3 } \cdot l $$$$ 60 = 12 \sqrt{ 3 } + 12 \sqrt{ 3 } \cdot l $$$$ 12 \sqrt{ 3 } \cdot l = 60 - 12 \sqrt{ 3 } $$$$ l = \frac{ 60 - 12 \sqrt{ 3 } }{ 12 \sqrt{ 3 } }$$$$ l = \frac{ 39.2154 }{ 12 \sqrt{ 3 } }$$$$ l = 1.8868 $$

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