Find the volume $ V $ of a triangular prism if side $a = 1$ and lateral $AL = \frac{\sqrt{ 3 }}{ 2 }$.

$V = \frac{ 1 }{ 4 }$

STEP 1: find length $ l $

To find length $ l $ use formula:

$$ AL = 3 \cdot a \cdot l $$

After substituting $ AL = \frac{\sqrt{ 3 }}{ 2 } $ and $ a = 1 $ we have:

$$ \frac{\sqrt{ 3 }}{ 2 } = 3 \cdot 1 \cdot l $$$$ \frac{\sqrt{ 3 }}{ 2 } = 3 \cdot l $$$$ l = \dfrac{ \frac{\sqrt{ 3 }}{ 2 } }{ 3 } $$$$ l = \frac{\sqrt{ 3 }}{ 6 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 1 $ and $ l = \frac{\sqrt{ 3 }}{ 6 } $ we have:

$$ V =\dfrac{ \sqrt{ 3 } \cdot 1 ^{ 2 } \cdot \frac{\sqrt{ 3 }}{ 6 } }{ 2 } $$$$ V = \dfrac{ \sqrt{ 3 } \cdot \frac{\sqrt{ 3 }}{ 6 } }{ 2 }$$$$ V = \dfrac{ \frac{ 1 }{ 2 } }{ 2 } $$$$ V = \frac{ 1 }{ 4 } $$

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