# The base of a triangular pyramid is a triangle with corners at (3 ,4 ), (2 ,7 ), and (3 ,6 ). If the pyramid has a height of 4 , what is the pyramid's volume?

Jun 24, 2017

The volume of the pyramid is $= \frac{4}{3} {u}^{3}$

#### Explanation:

The area of the base triangle is

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

$A = \frac{1}{2} | \left(3 , 4 , 1\right) , \left(2 , 7 , 1\right) , \left(3 , 6 , 1\right) |$

$= \frac{1}{2} \left(3 \cdot | \left(7 , 1\right) , \left(6 , 1\right) | - 4 \cdot | \left(2 , 1\right) , \left(3 , 1\right) | + 1 \cdot | \left(2 , 7\right) , \left(3 , 6\right) |\right)$

$= \frac{1}{2} \left(3 \left(7 - 6\right) - 4 \left(2 - 3\right) + 1 \left(12 - 21\right)\right)$

$= \frac{1}{2} \left(3 + 4 - 9\right)$

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

The height is $h = 4$

The volume of the pyramid is

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

$= \frac{1}{3} \cdot 1 \cdot 4 = \frac{4}{3}$