What fraction of the volume of a pyramid?

Question

what fraction of the volume of a pyramid must be cut off by a plane parallel to the base if the pyramid thus form has a lateral area equal to one-half of the lateral area of the original pyramid?

(pls. show your solutions)

(thanks...)

Answer 1

Simplyfing: The pyramid is 1m high, the base area is quadratic and 1m by 1m.

The lateral area of the cut is 0.5m², which is sqrt(0.5) by sqrt(0.5) metres. (~0.707 by 0.707 m)

So the height of the upper pyramid is 0.707 metres, with a 0.707 by 0.707 base.

The Volume of the upper pyramid is 1/3*0.5*0.707, compared to 1/3*1*1. So the fraction must be 0.5*sqrt(0.5), thats about 0.3535

or 35%.

The same is true for any kind of pyramid, with any base polygon and any height.