Find the area $ A $ of a triangular prism if height $h = 10$ and lateral $AL = 500$.

$A = 615.4701$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ h = 10 $ we have:

$$ 10 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 10 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 20 $$ $$ a = \dfrac{ 20 }{ \sqrt{ 3 } } $$ $$ a = \frac{ 20 \sqrt{ 3}}{ 3 } $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

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

After substituting $ a = \frac{ 20 \sqrt{ 3}}{ 3 } $ we have:

$$ B = \dfrac{ \sqrt{ 3 } \cdot \left( \frac{ 20 \sqrt{ 3}}{ 3 } \right)^2 }{ 4 } $$ $$ B = \dfrac{ \sqrt{ 3 } \cdot \frac{ 400 }{ 3 } }{ 4 }$$ $$ B = \dfrac{ \frac{ 400 \sqrt{ 3}}{ 3 } }{ 4 }$$ $$ B = \frac{ 100 \sqrt{ 3}}{ 3 } $$

STEP 3: find area $ A $

To find area $ A $ use formula:

$$ A = 2 B + AL $$

After substituting $ B = \frac{ 100 \sqrt{ 3}}{ 3 } $ and $ AL = 500 $ we have:

$$ A = 2 \cdot \frac{ 100 \sqrt{ 3}}{ 3 } + 500 $$ $$ A = \frac{ 200 \sqrt{ 3}}{ 3 } + 500 $$ $$ A = 615.4701 $$

Triangular Prism Calculator