2008 AMC 12B Problems/Problem 18
Problem
A pyramid has a square base and vertex
. The area of square
is
, and the areas of
and
are
and
, respectively. What is the volume of the pyramid?
Solution
Let be the height of the pyramid and
be the distance from
to
. The side length of the base is 14. The side lengths of
and
are
and
, respectively. We have a systems of equations through the Pythagorean Theorem:
.
Setting them equal to each other and simplifying gives
.
Therefore, , and the volume of the pyramid is
.