# The base of a triangular pyramid is a triangle with corners at (5 ,8 ), (6 ,7 ), and (2 ,3 ). If the pyramid has a height of 2 , what is the pyramid's volume?

May 21, 2018

color(maroon)(V = (1/3) * A_t * h = 2.65 cubic units

#### Explanation:

Given : $A \left(5 , 8\right) , B \left(6 , 7\right) , C \left(2 , 3\right)$, h = 2

Using distance formula we can calculate the lengths of sides a, b, c.

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$a = \sqrt{{\left(6 - 2\right)}^{2} + {\left(7 - 3\right)}^{2}} = \sqrt{32} = 5.66$

$b = \sqrt{{\left(5 - 2\right)}^{2} + {\left(8 - 3\right)}^{2}} = \sqrt{34} = 5.83$

$c = \sqrt{{\left(5 - 6\right)}^{2} + {\left(8 - 7\right)}^{2}} = \sqrt{2} = 1.41$

Semi perimeter of base triangle $s = \frac{a + b + c}{2}$

$s = \frac{5.66 + 5.83 + 1.41}{2} = \frac{12.9}{2} = 6.45$

Area of triangle ${A}_{t} = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

${A}_{t} = \sqrt{6.45 \cdot \left(6.45 - 5.66\right) \cdot \left(6.45 - 5.83\right) \cdot \left(6.45 - 1.41\right)}$

${A}_{t} = 3.97$

Volume of pyramid $V = \left(\frac{1}{3}\right) \cdot {A}_{t} \cdot h$

color(maroon)(V = (1/3) * 3.97 * 2 = 2.65# cubic units