Find the area $ A $ of a pyramid if height $h = 30$ and slant height $s = 32$.

$A = 1921.3477$

STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

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

After substituting $ h = 30 $ and $ s = 32 $ we have:

$$ 30^2 + \frac{ a^2 }{ 4 } = 32^2 $$ $$ \frac{ a^2 }{ 4 } = 32^2 - 30^2 $$ $$ \frac{ a^2 }{ 4 } = 1024 - 900 $$ $$ a^2 = 124 \cdot 4 $$ $$ a^2 = 496 $$ $$ a = \sqrt{ 496 } $$$$ a = 4 \sqrt{ 31 } $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 4 \sqrt{ 31 } $ we have:

$$ B = \left(4 \sqrt{ 31 }\right)^2 $$ $$ B = 496 $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 4 \sqrt{ 31 } $ and $ s = 32 $ we have:

$$ L = 8 \sqrt{ 31 } \cdot 32 $$$$ L = 256 \sqrt{ 31 } $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 496 $ and $ L = 256 \sqrt{ 31 } $ we have:

$$ A = 496 + 256 \sqrt{ 31 } $$ $$ A = 496 + 256 \sqrt{ 31 } $$ $$ A = 1921.3477 $$

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