The base of a triangular pyramid is a triangle with corners at #(3 ,1 )#, #(4 ,9 )#, and #(5 ,7 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(3  ,1  )#, #(4  ,9   )#, and #(5  ,7  )#.  If the pyramid has a height of #7  #, what is the pyramid's volume?

Sophia Alfred · Accepted Answer

#11.69 units^3#

#sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#:.=sqrt((5-3)^2+(7-1)^2)#

#:.=sqrt((2)^2+(6)^2)#

#:.=sqrt((4)+(36))#

#:.=sqrt(40)#

side a#=6.325#units

#:.=sqrt((4-3)^2+(9-1)^2)#

#:.=sqrt((1)^2+(8)^2)#

#:.=sqrt((1)+(64))#

#:.=sqrt(65)#

side b#=8.062#units

#:.=sqrt((5-4)^2+(7-9)^2)#

#:.=sqrt((1)^2+(2)^2)#

#:.=sqrt((1)+(4))#

#:.=sqrt(5)#

side #c=2.236#

Hero's recipe:

Area of #triangle=sqrt(s(s-a)(s-b)(s-c))#

where #s=(a+b+c)/2#

#:.s=(6.325+8.062+2.236)/2#

#:.s=16.623/2#

#:.s=8.312#

#:.=sqrt(8.312(8.312-6.325)(8.312-8.062)(8.312-2.236)))#

#:.=sqrt((8.312)(1.987)(0.25)(6.076))#

#:.=sqrt(25.08771894)#

Area of #triangle=5.01units^2#

Volume of triangular prism#=1/3AxxH#

#A=#triangular base and H= height of pyramid

#:.=1/3xx5.01xx7#

#:.=11.69 units^3#

Parker Allen · Answer

undefined To find the volume of the triangular pyramid, we can use the formula for the volume of a pyramid, which is one-third of the product of the base area and the height.

First, we need to find the area of the triangular base. We can use the formula for the area of a triangle given its coordinates.

Let the coordinates of the vertices of the triangle be $ (x_1, y_1), (x_2, y_2), (x_3, y_3) $. Then, the area of the triangle is given by:

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

Substituting the given coordinates, we can calculate the area of the base.

After finding the area of the base, which is the triangle, we multiply it by the height of the pyramid and then divide by 3 to get the volume of the pyramid.