Question

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?

Answer 1

Volume of pyramid `V_p =(1/3) A_t * h = color(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 = (1/2) [x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3]`

`A_t =(1/2) [(4*2) + (3*3) + (5*4) - (3*4) - (5*2) - (4*3)]`

`A_t = (1/2) [8+9+20-12-10-12] = 1.5`

Volume of pyramid `V_p =(1/3) A_t * h = (1/3) * 1.5 * 5 = 2.5`