# The base of a triangular pyramid is a triangle with corners at (2 ,7 ), (5 ,3 ), and (8 ,2 ). If the pyramid has a height of 2 , what is the pyramid's volume?

##### 1 Answer
Mar 26, 2018

$5.67$ cubic units

#### Explanation:

First Of All, Find the Area of the Triangular Base.

So, The Area of the Base of The Pyramid :-

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

[If The Coordiantes for the vertices of the triangles are $\left({x}_{1} , {y}_{1}\right) ,$

$\left({x}_{2} , {y}_{2}\right) \mathmr{and} \left({x}_{3} , {y}_{3}\right)$ respectively.]

$= \frac{1}{2} \left\{2 \left(3 - 2\right) + 5 \left(2 - 7\right) + 8 \left(7 - 3\right)\right\}$ sq. units

$= \frac{1}{2} \left\{2 - 25 + 40\right\}$ sq. units

$= \frac{1}{2} \left(17\right)$ sq. units

$= 8.5$ sq. units.

So, The Volume of The Triangular Pyramid

$= \frac{1}{3} \left(a h\right)$ [Where $a$ is the area of the base and $h$ is the height of the pyramid.]

$= \frac{1}{3} \left(8.5 \cdot 2\right)$ cubic units

$= \frac{17}{3}$ cubic units

$= 5.67$ cubic units (approx.)

Hope this helps.