The base of a triangular pyramid is a triangle with corners at #(1 ,2 )#, #(5 ,2 )#, and #(7 ,5 )#. 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 #(1  ,2  )#, #(5  ,2   )#, and #(7  ,5  )#.  If the pyramid has a height of #7  #, what is the pyramid's volume?

Savannah Aguilar · Accepted Answer

Volume #=14 " "#cubic units

We compute for the area of the base which is a triangle first Let #P_1(x_1, y_1)=(1, 2)# Let #P_2(x_2, y_2)=(5, 2)# Let #P_3(x_3, y_3)=(7, 5)# Use the formula to compute for the area of the triangular base Area #=1/2[(x_1,x_2,x_3,x_1),(y_1,y_2,y_3,y_1)]=1/2[(1,5,7,1),(2,2,5,2)]# Area #=1/2*(x_1*y_2+x_2*y_3+x_3*y_1-x_2*y_1-x_3*y_2-x_1*y_3)# Area #=1/2*(2+25+14-10-14-5)# Area #=6# Volume #=1/3*Area*height# Volume #=1/3*6*7# Volume #=14 " "#cubic units

Noah Atkinson · Answer

#14" units"^3#

Area of the base is #b/2xxa" "->" " 2xx3 = 6 " units"^2#

The volume of a triangular pyramid is:

#"base area" xx 1/3 xx "height"#

The height is given as 7

#color(blue)("base area" xx 1/3 xx "height " ->" " 6xx1/3xx7 =14 " units"^3#

Ella Armstrong · 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, calculate the area of the base triangle using the coordinates provided. Then, plug the base area and the given height into the formula to find the volume.