1983 AIME Problems/Problem 11
Problem
The solid shown has a square base of side length . The upper edge is parallel to the base and has length
. All edges have length
. Given that
, what is the volume of the solid?
Solution
First, we find the height of the figure by drawing a perpendicular from the midpoint of to
. The hypotenuse of the triangle is the median of equilateral triangle
one of the legs is
. We apply the pythagorean theorem to find that the height is equal to
.
Next, we complete the figure into a triangular prism, and find the area, which is .
Now, we subtract off the two extra pyramids that we included, whose combined area is .
Thus, our answer is .