# Divided Hexagon III +-- {.image} [[DividedHexagonIII.png:pic]] > The purple lines create two kites inside this regular hexagon. What's the missing area? =-- ## Solution by [[Similarity]] and [[regular hexagon|Properties of a Regular Hexagon]] +-- {.image} [[DividedHexagonIIILabelled.png:pic]] =-- With the points labelled as above, angles $D \hat{I} F$ and $H \hat{I} B$ are equal since they are [[vertically opposite]] and angles $F \hat{E} D$ and $B \hat{A} H$ are both interior angles in a [[regular hexagon]] so are equal. This means that the two kites are [[similar]]. As the [[area scale factor]] is $9$, the [[scale factor|length scale factor]] is $3$. The length of $A I$ is then a third of $E I$, so $I$ is three quarters of the way from $E$ to $A$ and so is level with $C G$. Also, the length of $A B$ is one third of that of $E F$, so triangle $A C I$ has area three times that of triangle $A B I$ and so triangle $A C G$ has area $3$. The full hexagon therefore has area $18$. The blue and red areas consist of $10$ of this, leaving $8$ for the two side regions. The yellow region therefore has area $4$.