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

Question

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

Evelyn Arboleda · Accepted Answer

The volume of the pyramid is #=4/3u^3#

The base triangle's area is

#A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

#A=1/2|(3,4,1),(2,7,1),(3,6,1)|#

#=1/2(3*|(7,1),(6,1)|-4*|(2,1),(3,1)|+1*|(2,7),(3,6)|)#

#=1/2(3(7-6)-4(2-3)+1(12-21))#

#=1/2(3+4-9)#

#=1/2|-2|=1#

The height is #h=4#

The pyramid's volume is

#V=1/3*A*h#

#=1/3*1*4=4/3#

Sadie Adams · Answer

undefined To find the volume of the triangular pyramid, you can use the formula: 
$$ V = \frac{1}{3} \times \text{base area} \times \text{height} $$

First, you need to find the area of the base triangle using its vertices. Then, you can calculate the volume using the given height.

Using the vertices (3,4), (2,7), and (3,6) for the base triangle, you can calculate the area using the formula for the area of a triangle given its vertices.

$$ \text{Area} = \frac{1}{2} \left| (x_1(y_2 - y_3)) + (x_2(y_3 - y_1)) + (x_3(y_1 - y_2)) \right| $$

Once you have the area of the base triangle, plug it into the volume formula along with the given height of the pyramid to find the volume.