Expert: Ahmed Salami - 8/28/2010

Question
hi there if you can help me out that would be a huge favor. im a student of engineering and ive come across this problem that involves volumes and pyramids and proving a theory.

What fraction of the volume of a pyramid must be cut off by a plane parallel to the base if the pyramid thus formed has a lateral area equal to one half of the lateral area of the original pyramid?
the answer on the book says 0.353555 but i dont know how to get to that conclusion

Answer
Hi Francis,
The volume of the pyramid is given by
V = BH/3
where B is the base area and H is the perpendicular height to the apex.
Let the volume of the cut-off pyramid be v such that
v = bh/3
where b is its base area and h is its perpendicular height.
But, we need
b = B/2
For the pyramids, by similarity
B/H² = b/h²
2b/H² = b/h²
H² = 2h²
H = h√2
h = H√2/2

Now,
v = (B/2)(H√2/2)/3
 = (BH/3)(√2/4)
 = V√2/4
v/V = √2/4 = 0.353555...

Regards