# The base of a triangular pyramid is a triangle with corners at (2 ,5 ), (6 ,6 ), and (2 ,8 ). If the pyramid has a height of 4 , what is the pyramid's volume?

Feb 26, 2018

Volume of pyramid color(blue)(V = (1/3) A_b * h = (1/3) 4.76 * 4 = 6.35 cu. units

#### Explanation:

Triangular pyramid

Given : base corners and height.

To find the volume using formula $V = \left(\frac{1}{3}\right) {A}_{b} \cdot h$

A_b = sqrt(s (s-a) s- b) (s - c)) where s is the semi perimeter of the the triangular base and a,b,c are the three sides of the base. h is the height of the pyramid.

$a = \sqrt{{\left(6 - 2\right)}^{2} + {\left(8 - 6\right)}^{2}} = 4.47$

$b = \sqrt{{\left(2 - 2\right)}^{2} + {\left(8 - 5\right)}^{2}} = 3$

$c = \sqrt{{\left(6 - 5\right)}^{2} + {\left(8 - 5\right)}^{2}} = 3.16$

$s = \frac{4.47 + 3 + 3.16}{2} = 5.32$

${A}_{b} = \sqrt{5.32 \cdot \left(5.32 - 4.47\right) \left(5.32 - 3\right) \left(5.32 - 3.16\right)} = 4.76$

Volume of pyramid color(blue)(V = (1/3) A_b * h = (1/3) 4.76 * 4 = 6.35 cu. units