How many equilateral triangles are there in a regular hexagon?

Steps to Solve

1. Identify the vertices of a hexagon

A regular hexagon has 6 vertices. We can label them as $A$, $B$, $C$, $D$, $E$, and $F$.

2. Choose a vertex as a starting point

To form an equilateral triangle, we can choose any vertex to start from. Let's say we start from vertex $A$.

3. Find possible configurations

From vertex $A$, we can choose two other vertices to form the equilateral triangle. The possible pairs of vertices that can be chosen, while ensuring that they maintain the equal lengths needed for an equilateral triangle, are:

• $B$ and $C$
• $C$ and $D$
• $D$ and $E$
• $E$ and $F$
• $F$ and $A$
• $A$ and $B$

4. Count the total distinct equilateral triangles

Each triangle formed using a different starting vertex will generate similar configurations. However, we must remember that those triangles will be congruent.

• Starting at $A$, the triangles form: $\triangle ABC$, $\triangle ACD$, $\triangle ADE$, $\triangle AEF$, and $\triangle AFB$.
• Similarly, starting from vertices $B$, $C$, $D$, $E$, and $F$, we find that every selection mirrors those from $A$.

Therefore, for each starting vertex of a hexagon, we can create triangles that are already counted.

5. Sum up unique triangles

The unique triangles are:

• $\triangle ABC$
• $\triangle ACD$
• $\triangle ADE$
• $\triangle AEF$
• $\triangle AFB$
• plus the triangles formed as we rotate around the hexagon.

Ultimately, 8 unique equilateral triangles can be formed with the vertices of a regular hexagon.