# The base of a triangular pyramid is a triangle with corners at (1 ,1 ), (3 ,6 ), and (4 ,5 ). If the pyramid has a height of 5 , what is the pyramid's volume?

Jul 12, 2017

The volume is $= 5.83 {u}^{3}$

#### Explanation:

The area of the base is

$A = \frac{1}{2} | \left({x}_{1} , {y}_{1} , 1\right) , \left({x}_{2} , {y}_{2} , 1\right) , \left({x}_{3} , {y}_{3} , 1\right) |$

$= \frac{1}{2} \left({x}_{1} \left({y}_{2} - {y}_{3}\right) - {y}_{1} \left({x}_{2} - {x}_{3}\right) + \left({x}_{2} {y}_{3} - {x}_{3} {y}_{2}\right)\right)$

$A = \frac{1}{2} | \left(1 , 1 , 1\right) , \left(3 , 6 , 1\right) , \left(4 , 5 , 1\right) |$

$= \frac{1}{2} \left(1 \cdot | \left(6 , 1\right) , \left(5 , 1\right) | - 1 \cdot | \left(3 , 1\right) , \left(4 , 1\right) | + 1 \cdot | \left(3 , 6\right) , \left(4 , 5\right) |\right)$

$= \frac{1}{2} \left(1 \left(6 - 5\right) - 1 \left(3 - 4\right) + 1 \left(15 - 24\right)\right)$

$= \frac{1}{2} \left(1 + 1 - 9\right)$

$= \frac{1}{2} | - 7 | = \frac{7}{2}$

The volume of the pyramid is

$V = \frac{1}{3} \cdot A \cdot h$

$= \frac{1}{3} \cdot \frac{7}{2} \cdot 5$

$= \frac{35}{6}$

$= 5.83$