The top of a square based pyramid with side length 12 cm is cut off. The cut is made parallel to the base. If the base of the smaller pyramid has a side length of 3 cm and the vertical height of the truncated pyramid is 6 cm what is the volume of the truncated pyramid.
The cut made at the top of the larger pyramid which has a side length of 12 cm is parallel to the base. Doing this gives a smaller pyramid of side length 3 cm. The height of the truncated pyramid is 6 cm.
The volume of a pyramid is equal to `V = (1/3)*B*h` where B is the area of the base and h is the vertical height of the pyramid.
If the height of the smaller pyramid is x, `x/3 = (6+x)/12`
=> 12x = 18 + 3x
=> 9x = 18
=> x = 2
The height of the original pyramid is 8 cm.
The volume of the original pyramid is `(1/3)*8*144 = 384` . The volume of the pyramid that is cut off is `(1/3)*2*9 = 6` . The volume of the truncated pyramid is 384 - 6 = 378.
The required volume of the truncated pyramid is 378 cm^3.