The base of a triangular pyramid is a triangle with corners at #(3 ,1 )#, #(2 ,7 )#, and #(3 ,6 )#. If the pyramid has a height of #4 #, what is the pyramid's volume?

1 Answer
Jul 18, 2018

Volume of Pyramid # ~~ 3.3333# cubic units.

Explanation:

Using distance formula and Heron’s formula we can calculate the base Area of the pyramid. Multiplying it by one third the height will give the volume.

Distance formula #d = sqrt((x_2-x_1)^2 + (y_2 - y_1)^2)#

Heron’s formula #A_t = sqrt(s * (s-a) * (s-b) * (s-c))#

Where s is the semiperemter of the triangle and a, b, c are the three sides of the base triangle.

Let #A (3,1), B (2,7) C (3,6)#

#a = sqrt((2-3)^2 + (7-6)^2) = 1.4142#

#b = sqrt((3-3)^2 + (6-1)^2) = 5#

#c = sqrt((3-2)^2 + (1-7)^2) = 6.0828#

Semiperemeter #s = (a+b+c)/2 = 12.497/2 = 6.2485#

#A_t = sqrt(6.2485 * (6.2485-1.4142) * (6.2485-5) * (6.2485-6.0828))#

#A_t ~~ 2.5 # sq. units

Volume of pyramid #V_p =(1/3) * 2.5 * 4 ~~ 3.3333# cubic units.