# The base of a triangular pyramid is a triangle with corners at (2 ,4 ), (3 ,2 ), and (5 ,5 ). If the pyramid has a height of 5 , what is the pyramid's volume?

May 17, 2018

Volume of triangular base pyramid color(violet)(V = 466.47 cub. units

#### Explanation:

$A \left(2 , 4\right) , B \left(3 , 2\right) , C \left(5 , 5\right)$

Using distance formula,

$\overline{A B} = c = \sqrt{{\left(3 - 2\right)}^{2} + {\left(2 - 4\right)}^{2}} = \sqrt{5} = 2.235$

$\overline{B C} = a = \sqrt{{\left(5 - 2\right)}^{2} + {\left(5 - 4\right)}^{2}} = \sqrt{10} = 3.162$

$\overline{A C} = b = \sqrt{{\left(3 - 5\right)}^{2} + {\left(2 - 5\right)}^{2}} = \sqrt{13} = 3.606$

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

Semi perimeter $s \frac{a - + b + c}{2} = \frac{2.235 + 3.162 + 3.606}{2} = 18$

${A}_{b} = \sqrt{18 \cdot 15.765 \cdot 16.838 \cdot 16.394} = 279.88$

Formula for volume of pyramid $V = \left(\frac{1}{3}\right) \cdot {A}_{b} \cdot h$

 color(violet)(V = (1/3) * 279.88 * 5 = 466.47 cub. units.