Find the area $ A $ of a triangular prism if length $l = 9$ and base area $B = 10$.

$A = 149.7518$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ B = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 }$$

After substituting $ B = 10 $ we have:

$$ 10 = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 }$$ $$ 10 \cdot 4 = \sqrt{ 3 } \cdot a^2 $$ $$ 40 = \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 40 }{ \sqrt{ 3 } } $$ $$ a^2 = \frac{ 40 \sqrt{ 3}}{ 3 } $$ $$ a = \sqrt{ \frac{ 40 \sqrt{ 3}}{ 3 } } $$$$ a \approx 4.8056 $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 4.8056 $ and $ l = 9 $ we have:

$$ A = \frac{ 4.8056^2 \sqrt{3}}{2} + 3 \cdot 4.8056 \cdot 9 $$$$ A = \frac{ 23.094 \sqrt{3}}{2} + 3 \cdot 43.2506 $$$$ A = \frac{ 40}{2} + 129.7518 $$$$ A = 20 + 129.7518 $$$$ A = 149.7518 $$

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