# The base of a triangular pyramid is a triangle with corners at (7 ,7 ), (5 ,3 ), and (8 ,4 ). If the pyramid has a height of 6 , what is the pyramid's volume?

Volume $V = 10 \text{ }$cubic units

#### Explanation:

Let ${P}_{1} \left(7 , 7\right)$, and ${P}_{2} \left(5 , 3\right)$,and ${P}_{3} \left(8 , 4\right)$

Compute the area of the base of the pyramid
$A = \frac{1}{2} \left[\begin{matrix}{x}_{1} & {x}_{2} & {x}_{3} & {x}_{1} \\ {y}_{1} & {y}_{2} & {y}_{3} & {y}_{1}\end{matrix}\right]$

$A = \frac{1}{2} \left[{x}_{1} {y}_{2} + {x}_{2} {y}_{3} + {x}_{3} {y}_{1} - {x}_{2} {y}_{1} - {x}_{3} {y}_{2} - {x}_{1} {y}_{3}\right]$

$A = \frac{1}{2} \left[\begin{matrix}7 & 5 & 8 & 7 \\ 7 & 3 & 4 & 7\end{matrix}\right]$

$A = \frac{1}{2} \left(7 \cdot 3 + 5 \cdot 4 + 8 \cdot 7 - 5 \cdot 7 - 8 \cdot 3 - 7 \cdot 4\right)$

$A = \frac{1}{2} \left(21 + 20 + 56 - 35 - 24 - 28\right)$

$A = \frac{1}{2} \left(97 - 87\right)$

$A = 5$

Volume $V = \frac{1}{3} \cdot A h = \frac{1}{3} \cdot 5 \cdot 6 = \frac{30}{3} = 10$

God bless....I hope the explanation is useful.