# The base of a triangular pyramid is a triangle with corners at (3 ,8 ), (4 ,9 ), and (2 ,6 ). If the pyramid has a height of 7 , what is the pyramid's volume?

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Jim G. Share
Feb 9, 2018

$\text{volume } = \frac{7}{6}$

#### Explanation:

$\text{the volume (V) of a pyramid is calculated using the formula}$

•color(white)(x)V=1/3xx"area of base "xx"height"

$\text{the area of the triangle can be found using}$

$\textcolor{w h i t e}{x} A = \frac{1}{2} | {x}_{1} \left({y}_{2} - {y}_{3}\right) + {x}_{2} \left({y}_{3} - {y}_{1}\right) + {x}_{3} \left({y}_{1} - {y}_{2}\right) |$

$\text{let } \left({x}_{1} , {y}_{1}\right) = \left(3 , 8\right) , \left({x}_{2} , {y}_{2}\right) = \left(4 , 9\right) , \left({x}_{3} , {y}_{3}\right) = \left(2 , 6\right)$

$\Rightarrow A = \frac{1}{2} | 3 \left(9 - 6\right) + 4 \left(6 - 8\right) + 2 \left(8 - 9\right) |$

$\textcolor{w h i t e}{\Rightarrow A} = \frac{1}{2} | 9 - 8 - 2 | = \frac{1}{2} \times 1 = \frac{1}{2}$

$\Rightarrow V = \frac{1}{3} \times \frac{1}{2} \times 7 = \frac{7}{6}$

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