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

Question

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

Mason Adams · Accepted Answer

Volume of the pyramid #color(blue)(= 8 cm^3#

Volume of a triangular pyramid v = (1/3) * base area * pyramid height. Pyramid height = 6 cm Coordinates of the triangular base #(6,2), (3,7), (4,8)#

Area of triangular base = #sqrt(s (s-a) (s-b) (s-c))# where a, b, c are the three sides of the triangular base and s is the semi perimeter of the base #s = (a+b+c)/2#

To determine the sides of a triangle:

#a = sqrt((3-6)^2 + (7-2)^2) = sqrt(9 +25) = 5.831#

#b =sqrt ((4-3)^2 + (8-7)^2) = sqrt 2 = 1.4142#

#c = sqrt((4-6)^2 + (8-2)^2) = sqrt40 = 6.3246#

#s = (5.831 + 1.4142 + 6.3246) / 2 = 6.7849#

#s-a = 6.7849 - 5.831 = 0.9539# #s-b = 6.7849 - 1.4142 = 5.3707# #s-c = 6.7849 - 6.3246 = 0.4603#

Area of base #= sqrt (6.7849*0.9539*5.3707*0.4603)# Area of triangular base #= 4 cm^2#

Volume of pyramid #= (1/3)*4*6 = 8 cm^3#

Theodore Abad · Answer

undefined To find the volume of the triangular pyramid, you can use the formula:

Volume = (1/3) * base area * height

First, calculate the area of the base triangle using the coordinates given:

Area = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Then, use the height given for the pyramid.

Finally, plug the values into the formula to find the volume.