Find the height $ h $ of a pyramid if slant height $s = 90.3$ and base diagonal $d = 20.4$.

$h = 90.0115$

STEP 1: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d = 20.4 $ we have:

$$ 20.4 = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 20.4 }{ \sqrt{ 2 } } $$ $$ a \approx 14.425 $$

STEP 2: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ a = 14.425 $ and $ s = 90.3 $ we have:

$$ h ^ {\,2} + \frac{ 14.425^2 }{ 4 } = 90.3^2 $$ $$ h ^ {\,2} + \frac{ 208.08 }{ 4 } = 90.3^2 $$ $$ h ^ {\,2} + 52.02 = 90.3^2 $$ $$ h ^ {\,2} = 8154.09 - 52.02 $$ $$ h ^ {\,2} = 8102.07 $$ $$ h = \sqrt{ 8102.07 } $$$$ h = 90.0115 $$

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