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

Dec 6, 2017

Volume of the pyramid color(blue)(= 8 cm^3

#### Explanation:

Volume of a triangular pyramid v = (1/3) * base area * pyramid height.
Pyramid height = 6 cm
Coordinates of the triangular base $\left(6 , 2\right) , \left(3 , 7\right) , \left(4 , 8\right)$

Area of triangular base = $\sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$
where a, b, c are the three sides of the triangular base and s is the semi perimeter of the base
$s = \frac{a + b + c}{2}$

To find triangle sides :

$a = \sqrt{{\left(3 - 6\right)}^{2} + {\left(7 - 2\right)}^{2}} = \sqrt{9 + 25} = 5.831$

$b = \sqrt{{\left(4 - 3\right)}^{2} + {\left(8 - 7\right)}^{2}} = \sqrt{2} = 1.4142$

$c = \sqrt{{\left(4 - 6\right)}^{2} + {\left(8 - 2\right)}^{2}} = \sqrt{40} = 6.3246$

$s = \frac{5.831 + 1.4142 + 6.3246}{2} = 6.7849$

$s - a = 6.7849 - 5.831 = 0.9539$
$s - b = 6.7849 - 1.4142 = 5.3707$
$s - c = 6.7849 - 6.3246 = 0.4603$

Area of base $= \sqrt{6.7849 \cdot 0.9539 \cdot 5.3707 \cdot 0.4603}$
Area of triangular base $= 4 c {m}^{2}$

Volume of pyramid $= \left(\frac{1}{3}\right) \cdot 4 \cdot 6 = 8 c {m}^{3}$