# Solution to the [[Triangles in a Dodecagon]] Puzzle +-- {.image} [[TrianglesinaDodecagon.png:pic]] > The difference between the orange and yellow areas is $2$. What's the total area of the regular dodecagon? =-- ## Solution by Regions in a [[Regular Hexagon]] and [[Equilateral Triangle]] +-- {.image} [[TrianglesinaDodecagonAnnotated.png:pic]] =-- Label the points as above, so $A D$ is a [[diameter]] of the dodecagon and $O$ is its centre. Joining every other vertex creates a [[regular hexagon]] and so triangle $O C E$ is an [[equilateral triangle]], meaning that $C E$ has the same length as $O C$ and hence as $O B$. The height of $C$ above $A D$ is therefore half of that of $B$ above it, so using the [[area of a triangle]], triangle $A C D$ has half the area of triangle $A B D$. Triangle $A B D$ consists of the orange region plus triangle $A F D$, and triangle $A C D$ consists of the yellow region plus triangle $A F D$. Since the orange region is $2$ more than the yellow, this means that the area of triangle $A B D$ is $2$ more than that of triangle $A C D$. Since triangle $A C D$ has half the area of triangle $A B D$, this means that triangle $A C D$ has area $2$. Then as $O$ is the [[midpoint]] of $A D$, triangle $O C D$ has half the area of $A C D$, namely $1$. The dodecagon is made up of $12$ triangles [[congruent]] to $O C D$ and so has total area $12$.