# Solution to the Overlapping Triangle and Hexagon Puzzle +-- {.image} [[OverlappingTriangleandHexagon.png:pic]] > One corner of the regular hexagon is on the midpoint of the triangle's base. What fraction of the total area is shaded? =-- ## Solution by Properties of [[Regular Hexagons]] and [[Equilateral Triangles]] Let $x$ be the length of one side of the hexagon. Using [[lengths in a regular hexagon]], the [[equilateral triangle]] has height $2 x$ and so has base $\frac{4}{\sqrt{3}} x$. Its area is therefore $\frac{4}{\sqrt{3}} x^2$. The two remaining parts of the hexagon each can be reassembled into an [[equilateral triangle]] with side length $x$, so each has area $\frac{\sqrt{3}}{4} x^2$. The total area is therefore: $$ \frac{4}{\sqrt{3}} x^2 + \frac{\sqrt{3}}{2} x^2 = \frac{11}{2\sqrt{3}} x^2 $$ The unshaded area consists of the hexagon without the two upper triangles, so has area equal to four equilateral triangles with side length $x$. Its area is therefore $\sqrt{3} x^2$. The shaded area is therefore $$ \frac{11}{2\sqrt{3}} x^2 - \sqrt{3} x^2 = \frac{5}{2\sqrt{3}} x^2 $$ The fraction that is shaded is then $\frac{5}{11}$.