The star of David consists of overlapping congruent equilateral triangles arranged so the overlap is a regular hexagon as in the figure below. If the triangles in this particular star each has area 24 sq. cm., then what is the area of the overlap?

## Solution

Inspecting the star of David below, we see that the hexagonal overlap consists of 6 equilateral triangles of side $x$. Now the area of each of the big equilateral triangles is 24 AND we can see that each consists of 9 equilateral triangles of side $x$. Therefore the area of the hexagon is six-ninths the area of one of the big equilateral triangles, i.e., 16.