"Published in Newark, California, USA"
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?
Solution:
To illustrate the problem, it is better to draw a pyramid with any base as follows
 |
| Photo by Math Principles in Everyday Life |
If B1 is parallel to B2, then the two pyramids are similar. We can relate their lateral areas with their altitudes using the principles of ratio and proportion as follows
where S1 and S2 are the lateral area of the two pyramids, respectively. If S2 = 2S1, then the above equation becomes
Take the square root on both sides of the equation, we have
We can relate also their volumes with their altitudes using the principles of ratio and proportion as follows
but
Finally, their volume ratio is
