Find the slant height $ s $ of a pyramid if height $h = 12$ and base diagonal $d = 10$.

$s = \frac{\sqrt{ 626 }}{ 2 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ d = \sqrt{ 2 } \cdot a $$

After substituting $ d = 10 $ we have:

$$ 10 = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 10 }{ \sqrt{ 2 } } $$ $$ a = 5 \sqrt{ 2 } $$

STEP 2: find slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

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

After substituting $ h = 12 $ and $ a = 5 \sqrt{ 2 } $ we have:

$$ 12^2 + \frac{ \left(5 \sqrt{ 2 }\right)^2 }{ 4 }= s^2 $$ $$ 144 + \frac{ 50 }{ 4 }= s^2 $$ $$ 144 + \frac{ 25 }{ 2 } = s^2 $$ $$ s^2 = \frac{ 313 }{ 2 } $$ $$ s = \sqrt{ \frac{ 313 }{ 2 } } $$$$ s = \frac{\sqrt{ 626 }}{ 2 } $$

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