# The base of a triangular pyramid is a triangle with corners at #(4 ,2 )#, #(3 ,6 )#, and #(7 ,5 )#. If the pyramid has a height of #8 #, what is the pyramid's volume?

##### 2 Answers

#### Explanation:

The triangle with vertices

The two smaller triangles are right angled triangles with legs of lengths

The larger isosceles right angled triangle has area

So the total area of the given triangle is:

#4^2 - 4 - 9/2 = 15/2#

The pyramid has volume:

#1/3 * "base" * "height" = 1/3 * 15/2 * 8 = 20#

#### Explanation:

The area of a triangle with vertices

#"Area" = 1/2 abs(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)#

(see https://socratic.org/s/aK6h7PjW)

So putting:

#{ ((x_1, y_1) = (4, 2)), ((x_2, y_2) = (3, 6)), ((x_3, y_3) = (7, 5)) :}#

we find that the base of our pyramid has area:

#1/2 abs((4)(6)+(3)(5)+(7)(2)-(4)(5)-(3)(2)-(7)(6))#

#=1/2 abs(24+15+14-20-6-42)#

#=1/2 abs(-15)#

#=15/2#

Then the volume of the pyramid is:

#1/3 xx "base" xx "height" = 1/3 * 15/2 * 8 = 20#