# The base of a triangular pyramid is a triangle with corners at (3 ,8 ), (4 ,9 ), and (5 ,6 ). If the pyramid has a height of 7 , what is the pyramid's volume?

Feb 16, 2018

Volume of pyramid

$V = \left(\frac{1}{3}\right) \left({A}_{t} \cdot h\right) \approx \textcolor{p u r p \le}{5.1746}$

#### Explanation:

Given : Triangle corners A (3,8), B(4,9), C(5,6), height of pyramid h = 7.

To find volume of pyramid.

Using distance formula let’s find the sides of the base triangle.

$\vec{A B} = \sqrt{{\left(4 - 3\right)}^{2} + {\left(9 - 8\right)}^{2}} = \sqrt{2} = 1.4142$

$\vec{B C} = \sqrt{{\left(5 - 4\right)}^{2} + {\left(6 - 9\right)}^{2}} = \sqrt{10} = 3.1623$

$\vec{C A} = \sqrt{{\left(5 - 3\right)}^{2} + {\left(6 - 8\right)}^{2}} = \sqrt{13} = 3.6056$

Semi perimeter of triangle base

s = (a + b + c) / 2 = (1.4142 + 3.1623 + 3.6056)/2 ~~ color(blue)(4.09#

Area of base triangle formula, given three sides is

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

$\implies \sqrt{4.09 \left(4.09 - 3.1623\right) \left(4.09 - 3.6056\right) \left(1.4142\right)} \approx \textcolor{g r e e n}{2.2177}$

Volume of pyramid

$V = \left(\frac{1}{3}\right) \left({A}_{t} \cdot h\right) = \left(\frac{1}{3}\right) \cdot 2.2177 \cdot 7 \approx \textcolor{p u r p \le}{5.1746}$