Notes
four regular polygons solution

Four Regular Polygons

Solution by Congruent Triangles

In the diagram above, the lengths of $A B$ and $A D$ are the same, as are the lengths of $A C$ and $A E$. Angles $B \hat{A} D$ and $C \hat{A} E$ are equal, so the angles $B \hat{A} C$ and $D hat{A} E$ are equal. This means that triangles $B A C$ and $D A E$ are congruent, and so $C B$ and $E D$ have the same length. Since the triangles are equilateral, they are therefore congruent and so the area of the red triangle is also $3$.

Solution by Transformation

The rotation that is centred on $A$ that takes $B$ to $D$ also takes $C$ to $E$, and so triangles $A C B$ and $A E D$ are congruent.