# The base of a triangular pyramid is a triangle with corners at (7 ,6 ), (4 ,1 ), and (3 ,2 ). If the pyramid has a height of 6 , what is the pyramid's volume?

Jan 27, 2018

color(brown)(V = 32 cubic units

#### Explanation:

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

a, b c are the lengths of three sides and s the semi perimeter of the triangle $s = \frac{a + b + c}{2}$

To find the three sides by applying the distance formula,

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

a = sqrt((3-4)^2 + (2-1)^2) = sqrt2 ~~ color(red)(1.4142

b = sqrt((3-7)^2 + (2-6)^2) = sqrt32 ~~ color(red)(5.6568

c = sqrt((4-7)^2 + (1-6)^2) = sqrt34 ~~ color(red)(5.831

Semi perimeter of triangle

$s = \frac{1.4142 + 5.6568 + 5.831}{2} = \textcolor{red}{6.451}$

Area of triangle base

$B = \sqrt{6.451 \left(6.451 - 1.4142\right) \left(6.451 - 5.6568\right) \left(6.451 - 5.831\right)}$

$\textcolor{g r e e n}{B \approx 16}$ sq units

Volume of triangle based pyramid $V = \left(\frac{1}{3}\right) B h$ where h is the height of the pyramid.

$\textcolor{b r o w n}{V = \left(\frac{1}{3}\right) 16 \cdot 6 = 32}$ cubic units