The base of a triangular pyramid is a triangle with corners at #(2 ,5 )#, #(6 ,5 )#, and #(7 ,8 )#. If the pyramid has a height of #15 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(2 ,5  )#, #(6  ,5   )#, and #(7  ,8  )#.  If the pyramid has a height of #15  #, what is the pyramid's volume?

Savannah Aguilar · Accepted Answer

#30 unit^3#

Volume of pyramic #= 1/3*basearea*height#

Base area is equal to 1/2x[2,6,7,2] [5,5,8, 5].

the area is calculated as follows using the Cartesian coordinates (a,b), (c,d), and (e,f): =1/2*{a, c, e, a][b, d, f, b] = 1/2[((axd)+(cxf)+(exb))-((bxc)+(dxe)+(fxa))]

#= 1/2[((2*5)+(6*8)+(7*5))-((5*6)+(5*7)+(8*2))]# #= 1/2[(10+48+35) - (30+35+16)]# #= 1/2(93-81)# # = 1/2*12# =6

#Base Area = 6 unit^2#

#Volume = 1/3*6*15# #Volume = 30 unit^3#

Carson Arnold · 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 use the formula for the area $ A $ of a triangle given its vertices $ (x_1, y_1), (x_2, y_2), $ and $ (x_3, y_3) $ as:

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

Once we have the base area $ A $, we can plug it into the formula for the volume of a pyramid along with the given height to find the volume $ V $.