Find the area $ A $ of a triangular prism if length $l = 24$ and lateral $AL = 500$.

$A = 541.7643$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ AL = 500 $ and $ l = 24 $ we have:

$$ 500 = 3 \cdot a \cdot 24 $$$$ 500 = 72 \cdot a $$$$ a = \dfrac{ 500}{ 72 } $$$$ a = \frac{ 125 }{ 18 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = \frac{ 125 }{ 18 } $ and $ l = 24 $ we have:

$$ A = \frac{ \frac{ 125 }{ 18 }^2 \sqrt{3}}{2} + 3 \cdot \frac{ 125 }{ 18 } \cdot 24 $$$$ A = \frac{ \frac{ 15625 }{ 324 } \sqrt{3}}{2} + 3 \cdot \frac{ 500 }{ 3 } $$$$ A = \frac{ \frac{ 15625 \sqrt{ 3}}{ 324 }}{2} + 500 $$$$ A = \frac{ 15625 \sqrt{ 3}}{ 648 } + 500 $$$$ A = 541.7643 $$

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