Find the volume $ V $ of a pyramid if height $h = 8$ and slant height $s = 12$.

$V = \frac{ 2560 }{ 3 }$

STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

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

After substituting $ h = 8 $ and $ s = 12 $ we have:

$$ 8^2 + \frac{ a^2 }{ 4 } = 12^2 $$ $$ \frac{ a^2 }{ 4 } = 12^2 - 8^2 $$ $$ \frac{ a^2 }{ 4 } = 144 - 64 $$ $$ a^2 = 80 \cdot 4 $$ $$ a^2 = 320 $$ $$ a = \sqrt{ 320 } $$$$ a = 8 \sqrt{ 5 } $$

STEP 2: find volume $ V $

To find volume $ V $ use formula:

$$ V = \dfrac{ a ^{ 2 } \cdot h }{ 3 } $$

After substituting $ a = 8 \sqrt{ 5 } $ and $ h = 8 $ we have:

$$ V = \dfrac{ 320 \cdot 8 }{ 3 }$$$$ V = \dfrac{ 2560 }{ 3 } $$$$ V = \frac{ 2560 }{ 3 } $$

Pyramid Calculator