The base of a triangular pyramid is a triangle with corners at #(5 ,1 )#, #(3 ,6 )#, and #(4 ,8 )#. 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 #(5  ,1  )#, #(3  ,6   )#, and #(4  ,8  )#.  If the pyramid has a height of #5  #, what is the pyramid's volume?

Avery Allen · Accepted Answer

Volume of pyramid is #7.5 # cubic.unit.

Volume of a pyramid is #1/3*#base area #*#hight. #(x_1,y_1)=(5,1) ,(x_2,y_2)=(3,6),(x_3,y_3)=(4,8) , h=5# Triangle's area is #A_b = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|# #A_b = |1/2(5(6−8)+3(8−1)+4(1−6))|# or #A_b = |1/2(-10+21-20)| = 1/2| -9| =9/2# sq.unit. Volume of pyramid is #V=1/3*A_b*h # or #V= 1/3*9/2*5 = 7.5 # cubic.unit [Ans]

Sadie Adams · Answer

undefined The volume of the triangular pyramid is 20 cubic units.

Carson Arnold · Answer

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

Volume = (1/3) * Area of Base * Height

First, calculate the area of the base triangle using the coordinates provided. You can use the Shoelace formula or another method to find the area.

Let's denote the coordinates of the vertices of the triangle as A(5, 1), B(3, 6), and C(4, 8).

Using the Shoelace formula:
Area = (1/2) * |(5*6 + 3*8 + 4*1) - (1*3 + 6*4 + 8*5)|

Area = (1/2) * |(30 + 24 + 4) - (3 + 24 + 40)|

Area = (1/2) * |(58) - (67)|

Area = (1/2) * |(-9)|

Area = 4.5 square units

Now, you have the area of the base triangle and the height of the pyramid, which is given as 5 units.

Substitute these values into the formula for volume:
Volume = (1/3) * 4.5 * 5

Volume = (1/3) * 22.5

Volume = 7.5 cubic units

Therefore, the volume of the triangular pyramid is 7.5 cubic units.