Find the volume $ V $ of a pyramid if side $a = 19$ and slant height $s = 22.5$.

$V = 2454.3281$

STEP 1: find height $ h $

To find height $ h $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $ a = 19 $ and $ s = 22.5 $ we have:

$$ h ^ {\,2} + \frac{ 19^2 }{ 4 } = 22.5^2 $$ $$ h ^ {\,2} + \frac{ 361 }{ 4 } = 22.5^2 $$ $$ h ^ {\,2} + \frac{ 361 }{ 4 } = 22.5^2 $$ $$ h ^ {\,2} = 506.25 - \frac{ 361 }{ 4 } $$ $$ h ^ {\,2} = 416 $$ $$ h = \sqrt{ 416 } $$$$ h = 20.3961 $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

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

After substituting $ a = 19 $ and $ h = 20.3961 $ we have:

$$ V = \dfrac{ 361 \cdot 20.3961 }{ 3 }$$$$ V = \dfrac{ 7362.9842 }{ 3 } $$$$ V = 2454.3281 $$

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