# The base of a triangular pyramid is a triangle with corners at (1 ,5 ), (6 ,2 ), and (5 ,9 ). If the pyramid has a height of 8 , what is the pyramid's volume?

Jul 2, 2017

$V = 42 \frac{2}{3}$ units.

#### Explanation:

The volume of a pyramid is $V = B \cdot h$, where $B$ is the area of the base, and $h$ is the height of the pyramid.

The area of the base can be found by subtracting triangles from a rectangle. The graph of the base is shown below.

We can subtract 3 triangles from the the rectangle.

The top left triangle has an area of $\frac{1}{2} \cdot 4 \cdot 4 = 8$.
The top right triangle has an area of $\frac{1}{2} \cdot 1 \cdot 7 = 3.5$.
The bottom left triangle has an area of $\frac{1}{2} \cdot 3 \cdot 5 = 7.5$.

The sum of the areas of these 3 triangles is $8 + 3.5 + 7.5 = 19$. The area of the rectangle is $5 \cdot 7 = 35$. So, the area of the base of the pyramid is $35 - 19 = 16$.

Plugging this into the formula for the volume, we have $V = \frac{1}{3} \cdot 16 \cdot 8 = \frac{128}{3} = 42 \frac{2}{3}$ units.