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?

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?

Jayden Jackson · Accepted Answer

Volume of pyramid #V_p =(1/3) A_t * h = color(red)(2.5)#

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#

Jayden Jackson · Answer

undefined The volume of a triangular pyramid is given by the formula V = (1/3) * base area * height.

To find the base area, we can use the formula for the area of a triangle given its vertices (x1, y1), (x2, y2), and (x3, y3):

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Using the given coordinates, we can find the base area as follows:

x1 = 4, y1 = 4
x2 = 3, y2 = 2
x3 = 5, y3 = 3

Area = 1/2 * |4(2 - 3) + 3(3 - 4) + 5(4 - 2)| = 1/2 * |-1 - 1 + 10| = 1/2 * 10 = 5

Now, we can calculate the volume of the pyramid:

V = (1/3) * base area * height = (1/3) * 5 * 5 = 25/3 = 8.33 (approximately)