Find the height $ h $ of a pyramid if total surface $A = 126.42$ and base area $B = 6.5$.

$h = 23.4837$

STEP 1: find side $ a $

To find side $ a $ use formula:

After substituting $ B = 6.5 $ we have:

$$ B = a^2 $$ $$ a^2 = 6.5 $$ $$ a = \sqrt{ 6.5 } $$$$ a \approx 2.5495 $$

STEP 2: find lateral surface $ L $

To find lateral surface $ L $ use formula:

$$ A = B + L $$

After substituting $ A = 126.42 $ and $ B = 6.5 $ we have:

$$ 126.42 = 6.5 + L $$ $$ L = 126.42 - 6.5 $$ $$ L = 119.92 $$

STEP 3: find slant height $ s $

To find slant height $ s $ use formula:

$$ L = 2 \cdot a \cdot s $$

After substituting $ L = 119.92 $ and $ a = 2.5495 $ we have:

$$ 119.92 = 2 \cdot 2.5495 \cdot s $$$$ 119.92 = 5.099 \cdot s $$$$ s = \dfrac{ 119.92 }{ 5.099 } $$$$ s = 23.5182 $$

STEP 4: find height $ h $

To find height $ h $ use Pythagorean Theorem:

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

After substituting $ a = 2.5495 $ and $ s = 23.5182 $ we have:

$$ h ^ {\,2} + \frac{ 2.5495^2 }{ 4 } = 23.5182^2 $$ $$ h ^ {\,2} + \frac{ 6.5 }{ 4 } = 23.5182^2 $$ $$ h ^ {\,2} + 1.625 = 23.5182^2 $$ $$ h ^ {\,2} = 553.1079 - 1.625 $$ $$ h ^ {\,2} = 551.4829 $$ $$ h = \sqrt{ 551.4829 } $$$$ h = 23.4837 $$

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