# The base of a triangular pyramid is a triangle with corners at (2 ,4 ), (3 ,2 ), and (8 ,5 ). If the pyramid has a height of 5 , what is the pyramid's volume?

Mar 1, 2016

$\frac{65}{6}$

#### Explanation:

The volume of a pyramid is a third of the base area multiplied by the height.

$V = \frac{1}{3} \times A \times h$

The easiest way to compute the base area is using the Heron's formula.

Using the Pythagorean Theorem, we can get the length of the 3 edges of the triangle.

${l}_{1} = \sqrt{{\left(2 - 3\right)}^{2} + {\left(4 - 2\right)}^{2}} = \sqrt{5}$

${l}_{2} = \sqrt{{\left(2 - 8\right)}^{2} + {\left(4 - 5\right)}^{2}} = \sqrt{37}$

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

The semi-perimeter is given by

$s = \frac{{l}_{1} + {l}_{2} + {l}_{3}}{2} \approx 7.07$

The area given by Heron's formula is

$A = \sqrt{s \left(s - {l}_{1}\right) \left(s - {l}_{2}\right) \left(s - {l}_{3}\right)} = \frac{13}{2}$

The volume of the pyramid is

$V = \frac{1}{3} \times \frac{13}{2} \times 5 = \frac{65}{6}$