Regular hexagon side and diagonal: how to find $AD$ and angle $ADF$

Expert intro: A regular hexagon is one of those shapes where symmetry does most of the heavy lifting. Once you spot the right triangles inside it, the rest is clean trig and geometry.

Detailed walkthrough

Step 1: Use the structure of a regular hexagon

A regular hexagon can be split into 6 equilateral triangles by joining the center to each vertex. That means each central angle is

$\frac{360^\circ}{6}=60^\circ.$

Also, the side length is $5.5$ cm.

Step 2: Find the diagonal $AD$

Vertices $A$ and $D$ are opposite each other, so $AD$ passes through the center of the hexagon. In a regular hexagon, the distance from the center to a vertex equals the side length.

So

$AD = 2\times 5.5 = 11\text{ cm}.$

Step 3: Find $\angle ADF$

Now look at triangle $ADF$.

• $AD$ is a long diagonal through the center
• $DF$ is a side of the hexagon
• the angle at $D$ can be found using the isosceles/equilateral structure inside the hexagon

A cleaner way is to use coordinates or the fact that the hexagon is made from $60^\circ$ steps. The direction from $D$ to $A$ and from $D$ to $F$ forms an angle of

$\angle ADF = 30^\circ.$

Step 4: Final answers

• $AD=11\text{ cm}$
• $\angle ADF=30^\circ$

💡 Pitfall guide

The big trap is treating the diagonal as if it were just another side of a rhombus. It is not. In a regular hexagon, the opposite-vertex diagonal is twice the side length, and that shortcut saves a lot of time. Also, don’t force cosine rule too early if symmetry gives you the angle faster.

🔄 Real-world variant

If the question asked for a shorter diagonal like $AC$ or $DF$, those lengths are different: $DF$ is just a side, while $AC$ is the diagonal of one of the equilateral triangles’ combined shapes. The same symmetry idea still works, but the answer is not always the full diameter-like diagonal $AD$.

🔍 Related terms

regular hexagon, symmetry, central angle