What Is a Diagonal of a Hexagon?
A diagonal of a hexagon is a straight line segment that joins two vertices (corners) that are not next to each other. A hexagon has six vertices; the lines joining adjacent corners are the sides, and every other corner-to-corner line is a diagonal.
From any one corner you cannot draw a diagonal to itself or to its two neighbours (those are sides). That leaves three corners you can reach, so each vertex sends out exactly three diagonals.
How Many Diagonals Does a Hexagon Have?
This is the question readers ask most, and it has a clean answer: a hexagon has 9 diagonals. The number comes from the general rule for any polygon with $n$ sides:
$$\text{Number of diagonals} = \frac{n(n-3)}{2}$$
Substituting $n = 6$ for a hexagon:
$$\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9$$
Why does the formula work? Each of the $n$ vertices can connect to $(n-3)$ others by a diagonal - every vertex except itself and its two neighbours. That gives $n(n-3)$ endpoints, but each diagonal has been counted twice (once from each end), so you divide by $2$. This rule holds for every simple polygon, not just the hexagon.
How Long Are the Diagonals of a Regular Hexagon?
In a regular hexagon with side length $s$, the nine diagonals are not all the same length. They come in two sizes.
The 3 long diagonals run from a corner straight through the centre to the opposite corner. Each one is exactly two side-lengths long:
$$\text{Long diagonal} = 2s$$
The 6 short diagonals skip one corner and do not pass through the centre. Each connects two vertices that have one vertex between them:
$$\text{Short diagonal} = \sqrt{3},\cdot s$$
The long diagonal is a diameter of the circle that passes through all six corners; that is why it equals $2s$, since a regular hexagon's side equals the radius of that circle. The short diagonal is the base of an equilateral-triangle relationship, which is where the $\sqrt{3}$ comes from.
Examples of Diagonal of Hexagon
These examples build from counting diagonals to computing their lengths. Each problem statement is bold; the steps are plain.
Example 1
How many diagonals does a hexagon have? Use the polygon formula.
Use $\dfrac{n(n-3)}{2}$ with $n = 6$.
$$\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9$$
Final answer: a hexagon has $9$ diagonals.
Example 2
A student tries to count a hexagon's diagonals by hand and gets $18$. Where did it go wrong, and what is the right count?
The intuitive method is to visit each of the six corners, count the three diagonals leaving it, and add them: $6 \times 3 = 18$. Let's check that against a drawing.
Take diagonal $AD$. Counting from corner $A$ you record it once, and then counting from corner $D$ you record the very same line again. Every diagonal has two ends, so this method counts each one twice.
Divide the raw total by $2$ to remove the double count:
$$\frac{18}{2} = 9$$
Final answer: $9$ diagonals. The $18$ was not wrong arithmetic; it was the un-halved count, which is exactly why the formula ends in $\div 2$.
Example 3
A regular hexagon has a side of $5$ cm. Find the length of a long diagonal.
A long diagonal passes through the centre and equals two sides.
$$\text{Long diagonal} = 2s = 2 \times 5 = 10 \text{ cm}$$
Final answer: the long diagonal is $10$ cm.
Example 4
The same regular hexagon has a side of $5$ cm. Find the length of a short diagonal.
A short diagonal equals $\sqrt{3}$ times the side.
$$\text{Short diagonal} = \sqrt{3},\cdot s = \sqrt{3} \times 5 \approx 1.732 \times 5 = 8.66 \text{ cm}$$
Final answer: the short diagonal is $5\sqrt{3} \approx 8.66$ cm, shorter than the long diagonal, as expected. A frequent slip is to assume all diagonals are equal and quote one length for all nine.
Example 5
A regular hexagon has a long diagonal of $14$ cm. What is its side length?
The long diagonal is $2s$, so work backwards.
$$2s = 14 \implies s = \frac{14}{2} = 7 \text{ cm}$$
Final answer: the side length is $7$ cm.
Example 6
How many of a hexagon's diagonals pass through its centre, and how many do not?
In a regular hexagon, only the diagonals joining opposite corners pass through the centre. There are three such pairs of opposite corners, so $3$ long diagonals pass through the centre.
The remaining diagonals are $9 - 3 = 6$, the short ones, which miss the centre.
Final answer: $3$ diagonals pass through the centre; $6$ do not.
Where Do Hexagon Diagonals Actually Matter?
Hexagons are everywhere in engineering and nature because they tile a plane with no gaps, and their diagonals carry real structural meaning.
Bracing and trusses. In hexagonal frameworks, the long diagonals act as struts that stiffen the shape against collapse. A plain hexagon of hinged rods folds flat; add the diagonals and it holds its form. The count and length of those diagonals decide how much material a design needs.
Honeycomb and materials. Bees build hexagonal cells because the shape stores the most area for the least wall. Engineers copy this in honeycomb panels for aircraft floors, where the geometry of the hexagon, diagonals included, gives strength at very low weight.
Nuts, bolts, and tiling. A spanner grips a hex nut across a short diagonal, which is why the "size" of a nut is quoted across the flats and relates to that $\sqrt{3},s$ measurement.
The reason a hexagon tiles so efficiently, and why its diagonals lock a frame in place, is the same geometry that appears in honeycomb structures used from beehives to spacecraft panels. The diagonals are not decoration; they are the load paths.
What Are Common Mistakes With the Diagonal of a Hexagon?
These errors show up whenever a reader stops trusting the formula and starts counting or assuming.
Mistake 1: Forgetting to divide by 2
Where it slips in: Using $n(n-3)$ as the whole formula and stopping there.
Don't do this: Reporting $6 \times 3 = 18$ diagonals.
The correct way: The full formula is $\dfrac{n(n-3)}{2}$. Each diagonal is shared by two vertices, so the raw product double-counts every line. The rusher who writes the numerator and reads it off before dividing lands on $18$ every time.
Mistake 2: Counting the sides as diagonals
Where it slips in: Drawing every line between corners and calling them all diagonals.
Don't do this: Including the six sides in the diagonal count and quoting $15$.
The correct way: A diagonal joins non-adjacent corners only. The lines between neighbouring corners are sides, not diagonals. Subtract the sides before you count, or use the formula, which already excludes them.
Mistake 3: Assuming all diagonals have the same length
Where it slips in: Treating "the diagonal of a hexagon" as a single number.
Don't do this: Quoting $2s$ for every diagonal.
The correct way: A regular hexagon has two diagonal lengths: long ones ($2s$) through the centre and short ones ($\sqrt{3},s$) that skip a corner. The confusion here is the same one that trips students on a rectangle, where the two diagonals are equal but the label "diagonal" still hides more than one segment.
Conclusion
A hexagon has 9 diagonals, found from $\dfrac{n(n-3)}{2}$ with $n = 6$.
The formula divides by $2$ because each diagonal is counted from both of its endpoints.
A regular hexagon has two diagonal lengths: $3$ long diagonals of $2s$ through the centre and $6$ short diagonals of $\sqrt{3},s$.
Diagonals join non-adjacent corners only; the six sides are not diagonals.
Not every diagonal of a hexagon is the same length, unlike the two equal diagonals of a rectangle.
To work through polygon diagonals with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor, or structured math classes online.
Practise What You Have Learned
Work through these to test your understanding: how many diagonals does a pentagon have (Answer to Question 1: $5$, from $\dfrac{5 \times 2}{2}$); find the long diagonal of a regular hexagon with side $9$ cm (Answer to Question 2: $18$ cm); and find the short diagonal of the same hexagon (Answer to Question 3: $9\sqrt{3} \approx 15.59$ cm). To see the diagonals appear one by one as a hexagon is built, book a free demo class.
Read More
Hexagonal Prism — how the six-sided base extends into a solid.
Types of Polygon — where the hexagon sits among regular and irregular shapes.
Diagonal of Rhombus — a four-sided shape whose two diagonals meet at right angles.
Sum of Angles in a Polygon — the companion formula for interior angles.
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