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The two pieces of pyramid are rather complicated, but the bottom piece is easy to deal with. Consider the two parallel plane sections through E and F, perpendicular to the base and segment EF. This cuts the bottom piece into three pieces. The middle piece is a (sideways) prism. The other two pieces can be pushed together to make a new pyramid with a rectangular base.
By similar triangles (seen in side view of the original pyramid). If EF=x, the height of the rectangle based pyramid is (1-x) and the new width C'D' is also (1-x).
The prism volume is then (1/2)(1-x)(x)
The pyramid volume is (1/3)(1-x)(1-x)
For a total volume -x^2/6 - x/6 + 1/3
Equate this to half the original pyramid (1/3)/2 = 1/6
and simplify to the quadratic x^2 + x + 1 = 0
The solution is
(sqrt(5)-1)/2 which is the reciprocal of the golden ratio. |
