The base of a triangular pyramid is a triangle with corners at #(2 ,4 )#, #(3 ,2 )#, and #(5 ,5 )#. If the pyramid has a height of #5 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(2  ,4  )#, #(3  ,2   )#, and #(5  ,5  )#.  If the pyramid has a height of #5  #, what is the pyramid's volume?

Savannah Aguilar · Accepted Answer

Volume of triangular base pyramid #color(violet)(V = 466.47# cub. units

#A (2,4), B (3,2), C (5,5)# Applying the distance formula, #bar (AB) = c = sqrt((3-2)^2 + (2-4)^2) = sqrt5 = 2.235# #bar (BC) = a = sqrt((5-2)^2 + (5-4)^2) = sqrt 10 = 3.162# #bar (AC) = b = sqrt((3-5)^2 + (2-5)^2) = sqrt 13= 3.606# Area of base triangle #A_b= sqrt(s (s-a) (s-b) (s-c))# Semi perimeter #s (a -+ b+ c) / 2 = (2.235 + 3.162 + 3.606)/2 = 18# #A_b = sqrt(18* 15.765 * 16.838 * 16.394) = 279.88# Formula for volume of pyramid #V = (1/3) * A_b * h# # color(violet)(V = (1/3) * 279.88 * 5 = 466.47# cub. units.

Autumn Armstrong · Answer

undefined To find the volume of the triangular pyramid, first, calculate the area of the base triangle using the given coordinates of its vertices and then use the formula for the volume of a pyramid.

Step 1: Find the area of the base triangle using the coordinates of its vertices.
Step 2: Once you have the area of the base triangle, use the formula for the volume of a pyramid: Volume = (1/3) * base area * height.

After calculating the area of the base triangle and the given height of the pyramid, plug these values into the formula to find the volume of the pyramid.