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

Jan 8, 2018

Volume of pyramid ${V}_{p} = \left(\frac{1}{3}\right) {A}_{t} \cdot h = \textcolor{red}{2.5}$

#### Explanation:

The shoelace formula or shoelace algorithm (also known as Gauss's area formula and the surveyor's formula[1]) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane.

Using shoelace formula to find the area of the given triangle base :

${A}_{t} = \left(\frac{1}{2}\right) \left[{x}_{1} {y}_{2} + {x}_{2} {y}_{3} + {x}_{3} {y}_{1} - {x}_{2} {y}_{1} - {x}_{3} {y}_{2} - {x}_{1} {y}_{3}\right]$

${A}_{t} = \left(\frac{1}{2}\right) \left[\left(4 \cdot 2\right) + \left(3 \cdot 3\right) + \left(5 \cdot 4\right) - \left(3 \cdot 4\right) - \left(5 \cdot 2\right) - \left(4 \cdot 3\right)\right]$

${A}_{t} = \left(\frac{1}{2}\right) \left[8 + 9 + 20 - 12 - 10 - 12\right] = 1.5$

Volume of pyramid ${V}_{p} = \left(\frac{1}{3}\right) {A}_{t} \cdot h = \left(\frac{1}{3}\right) \cdot 1.5 \cdot 5 = 2.5$