Find the area $ A $ of a triangular prism if length $l = 23$ and height $h = 9$.

$A = 468 \sqrt{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ h = 9 $ we have:

$$ 9 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 9 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 18 $$ $$ a = \dfrac{ 18 }{ \sqrt{ 3 } } $$ $$ a = 6 \sqrt{ 3 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ a = 6 \sqrt{ 3 } $ and $ l = 23 $ we have:

$$ A = \frac{ 6 \sqrt{ 3 }^2 \sqrt{3}}{2} + 3 \cdot 6 \sqrt{ 3 } \cdot 23 $$$$ A = \frac{ 108 \sqrt{3}}{2} + 3 \cdot 138 \sqrt{ 3 } $$$$ A = \frac{ 108 \sqrt{ 3 }}{2} + 414 \sqrt{ 3 } $$$$ A = 54 \sqrt{ 3 } + 414 \sqrt{ 3 } $$$$ A = 468 \sqrt{ 3 } $$

Triangular Prism Calculator