The base of a triangular pyramid is a triangle with corners at #(2 ,6 )#, #(5 ,2 )#, and #(8 ,7 )#. If the pyramid has a height of #18 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(2 ,6  )#, #(5  ,2   )#, and #(8  ,7  )#.  If the pyramid has a height of #18  #, what is the pyramid's volume?

Aaliyah Anderson · Accepted Answer

Volume of pyramid # = color (blue)(81.03)# #cm^3#

Volume of a triangular pyramid v = (1/3) * base area * pyramid height. Pyramid height = 18 cm Coordinates of the triangular base #(2,6), (5,2), (8,7)#

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((5-2)^2 + (2-6)^2) = sqrt(9 +16) = 5#

#b =sqrt ((8-5)^2 + (7-2)^2) = sqrt 34 = 5.831#

#c = sqrt((7-6)^2 + (8-2)^2) = sqrt37 = 6.0828#

#s = (5 + 5.831 + 6.0828) / 2 = 8.4569#

#s-a = 8.4579 - 5 = 3.4579# #s-b = 8.4579 - 5.831 = 2.6259# #s-c = 8.4579 - 6.0828 = 2.3751#

Area of base #= sqrt (8.4569*3.4579*2.6259*2.3751)# Area of triangular base #= 13.505 cm^2#

Volume of pyramid #= (1/3)*13.505*18 = 81.03 cm^3#

Aurora Ayala · Answer

undefined To find the volume of a triangular pyramid, we use the formula:

$$ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$

First, we need to find the area of the base triangle. We can do this using the formula for the area of a triangle given its vertices (x1, y1), (x2, y2), and (x3, y3):

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

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

Then, we plug the base area and the height of the pyramid into the formula for volume to find the volume of the pyramid. Make sure to use the appropriate units for consistency.