Find the area $ A $ of a triangular prism if length $l = 19$ and volume $V = 4495.4$.

$A = 1178.7432$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ V = 4495.4 $ and $ l = 19 $ we have:

$$ 4495.4 = \dfrac{ \sqrt{ 3 } \cdot a ^{ 2 } \cdot 19 }{ 2 } $$$$ 4495.4 \cdot 2 = \sqrt{ 3 } \cdot a ^{ 2 } \cdot 19 $$$$ 8990.8 = \sqrt{ 3 } \cdot a ^{ 2 } \cdot 19 $$$$ 8990.8 = 19 \sqrt{ 3 } \cdot a ^{ 2 } $$$$ a ^{ 2 } = \dfrac{ 8990.8}{ 19 \sqrt{ 3 } } $$$$ a ^{ 2 } = 273.2021 $$$$ a = \sqrt{ 273.2021 } $$$$ a = 16.5288 $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 16.5288 $ and $ l = 19 $ we have:

$$ A = \frac{ 16.5288^2 \sqrt{3}}{2} + 3 \cdot 16.5288 \cdot 19 $$$$ A = \frac{ 273.2021 \sqrt{3}}{2} + 3 \cdot 314.0477 $$$$ A = \frac{ 473.2}{2} + 942.1432 $$$$ A = 236.6 + 942.1432 $$$$ A = 1178.7432 $$

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