Find the area $ A $ of a pyramid if side $a = 75$ and height $h = 48$.

$A = 14761.7732$

STEP 1: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 75 $ we have:

$$ B = 75^2 $$ $$ B = 5625 $$

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 = 48 $ and $ a = 75 $ we have:

$$ 48^2 + \frac{ 75^2 }{ 4 }= s^2 $$ $$ 2304 + \frac{ 5625 }{ 4 }= s^2 $$ $$ 2304 + \frac{ 5625 }{ 4 } = s^2 $$ $$ s^2 = \frac{ 14841 }{ 4 } $$ $$ s = \sqrt{ \frac{ 14841 }{ 4 } } $$$$ s = \frac{ 3 \sqrt{ 1649}}{ 2 } $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

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

After substituting $ a = 75 $ and $ s = \frac{ 3 \sqrt{ 1649}}{ 2 } $ we have:

$$ L = 150 \cdot \frac{ 3 \sqrt{ 1649}}{ 2 } $$$$ L = 225 \sqrt{ 1649 } $$

STEP 4: find total surface $ A $

To find total surface $ A $ use formula:

$$ A = B + L $$

After substituting $ B = 5625 $ and $ L = 225 \sqrt{ 1649 } $ we have:

$$ A = 5625 + 225 \sqrt{ 1649 } $$ $$ A = 5625 + 225 \sqrt{ 1649 } $$ $$ A = 14761.7732 $$

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