Nonagon: Sides, Angles, and Diagonals of a 9-Gon

The Building That Chose Nine Sides on Purpose

The US Steel Tower in Pittsburgh is not a rectangle or a hexagon; from above it reads as a nine-sided figure, an irregular nonagon, with triangular steel columns at its corners. The choice was structural: nine faces let the architects wrap more usable floor around a strong central core than four faces would. Most people never count the sides, but the geometry that holds the tower up is nonagon geometry, and it starts with one question, how do nine sides share $1260^\circ$?

What Is a Nonagon?

A nonagon is a closed, flat figure bounded by nine straight sides. It is a polygon (a many-sided shape), and like every polygon it has as many vertices and angles as it has sides, so a nonagon has 9 sides, 9 vertices, and 9 interior angles. The name comes from the Latin nonus (ninth) and the Greek gonia (angle); the purely Greek name enneagon means the same shape.

A nonagon can be drawn many ways, tall, squashed, or perfectly even, but every version shares those nine sides. The differences between versions are captured by its types, covered below.

What Are the Angles of a Nonagon?

A nonagon's angles come in two families, interior and exterior, and each has a clean rule.

Interior angle sum. For any polygon with $n$ sides, drawing all diagonals from one vertex splits it into $(n - 2)$ triangles, each contributing $180^\circ$:

$$\text{Sum of interior angles} = (n - 2) \times 180^\circ.$$

For a nonagon, $n = 9$, so the sum is:

$$(9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ.$$

This $1260^\circ$ holds for every nonagon, regular or lopsided, because the nine angles always fill the same seven triangles. The general rule lives on interior angles and sum of angles in a polygon.

Each interior angle of a regular nonagon. When all nine angles are equal, divide the total evenly:

$$\frac{1260^\circ}{9} = 140^\circ.$$

Exterior angles. The exterior angles of any polygon sum to $360^\circ$. For a regular nonagon each one is:

$$\frac{360^\circ}{9} = 40^\circ.$$

A quick check confirms the pair fits together: interior $+$ exterior $= 140^\circ + 40^\circ = 180^\circ$, a straight line, exactly as it should be. The broader rule is on exterior angles of a polygon.

How Many Diagonals Does a Nonagon Have?

A diagonal joins two vertices that are not already connected by a side. The count for any polygon is:

$$\text{Number of diagonals} = \frac{n(n - 3)}{2}.$$

The logic: each of the $n$ vertices can connect to $(n - 3)$ others (every vertex except itself and its two neighbours), and dividing by $2$ removes the double-counting of each diagonal from both ends. For a nonagon:

$$\frac{9 \times (9 - 3)}{2} = \frac{9 \times 6}{2} = \frac{54}{2} = 27.$$

A nonagon has 27 diagonals, far more than a square's two, because the number grows quickly with sides. For a refresher on what counts as a diagonal at all, see what is a diagonal.

What Are the Types of Nonagon?

Nonagons split along two independent lines, so a single nonagon carries one label from each pair.

• Regular vs irregular. A regular nonagon has all nine sides equal and all nine angles equal to $140^\circ$. An irregular nonagon has sides or angles of differing measures, yet its interior angles still sum to $1260^\circ$.

• Convex vs concave. A convex nonagon has every interior angle less than $180^\circ$, so all vertices point outward. A concave nonagon has at least one interior angle greater than $180^\circ$, creating a "dent" where one vertex points inward.

A regular nonagon is always convex, but an irregular one can be either convex or concave. The general families sit on types of polygon, convex polygon, and concave polygons.

Where Do Nonagons Appear in the Real World?

Nine sides are unusual enough that spotting a nonagon feels deliberate. Beyond the US Steel Tower, the shape turns up in some coin designs, in decorative tiles and stained-glass windows, and in the Bahá'í tradition, where the nine-sided Lotus Temple layout and other Bahá'í houses of worship use nine sides and nine entrances to symbolise unity. In each case the nonagon is chosen for meaning or for how efficiently nine faces enclose space, not by accident.

Examples of a Nonagon

The examples begin with the angle sum and end with recovering the number of sides from a diagonal count.

Example 1

Find the sum of the interior angles of a nonagon.

$$(n - 2) \times 180^\circ = (9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ.$$

Final answer: $1260^\circ$.

Example 2

A student wants each interior angle of a regular nonagon and computes $\frac{360^\circ}{9} = 40^\circ$. Is that right?

Wrong attempt. The student divides $360^\circ$ by $9$ and reports $40^\circ$ as the interior angle.

Why it breaks. $\frac{360^\circ}{9}$ is the exterior angle, because it is the $360^\circ$ full turn shared out. An interior angle of $40^\circ$ would make the nonagon a spiky star, not the near-round shape a regular nonagon is; a glance at the figure shows each corner is clearly obtuse, not sharp.

Correct. Interior angles come from the interior sum:

$$\frac{1260^\circ}{9} = 140^\circ.$$

Final answer: each interior angle is $140^\circ$; the $40^\circ$ was the exterior angle.

Example 3

Find each exterior angle of a regular nonagon.

Exterior angles sum to $360^\circ$ and there are nine equal ones:

$$\frac{360^\circ}{9} = 40^\circ.$$

Final answer: $40^\circ$.

Example 4

How many diagonals does a nonagon have?

$$\frac{n(n - 3)}{2} = \frac{9 \times 6}{2} = 27.$$

Final answer: $27$ diagonals.

Example 5

In an irregular nonagon, eight of the interior angles add to $1130^\circ$. Find the ninth angle.

The nine angles must total $1260^\circ$, so:

$$1260^\circ - 1130^\circ = 130^\circ.$$

Final answer: the ninth angle is $130^\circ$.

Example 6

A regular polygon has each interior angle equal to $140^\circ$. How many sides does it have, and what is the polygon?

Each exterior angle is $180^\circ - 140^\circ = 40^\circ$, and the sides are $\frac{360^\circ}{40^\circ}$:

$$n = \frac{360^\circ}{40^\circ} = 9.$$

Final answer: $9$ sides, so it is a nonagon.

Where Do Students Trip Up on Nonagons?

The errors are almost all about which "$360$-or-$1260$" rule applies where.

Mistake 1: Using $\frac{360^\circ}{9}$ for the interior angle

Where it slips in: finding each angle of a regular nonagon.

Don't do this: divide $360^\circ$ by $9$ and call the $40^\circ$ result an interior angle.

The correct way: $\frac{360^\circ}{9}$ is the exterior angle. The interior angle comes from $\frac{1260^\circ}{9} = 140^\circ$. The habit that fixes this is asking "does my answer make the corner look sharp or blunt?", a regular nonagon's corners are blunt ($140^\circ$), so a $40^\circ$ answer is a red flag.

Mistake 2: Miscounting diagonals by including the sides

Where it slips in: the diagonal count.

Don't do this: connect every vertex to every other and get $\frac{9 \times 8}{2} = 36$, which counts the nine sides as if they were diagonals.

The correct way: a diagonal skips the two neighbouring vertices, so each vertex reaches $9 - 3 = 6$ others, giving $\frac{9 \times 6}{2} = 27$. The $-3$ (not $-1$) is what excludes the vertex itself and its two adjacent sides.

Mistake 3: Assuming every nonagon is regular

Where it slips in: applying $140^\circ$ to any nonagon.

Don't do this: claim each angle is $140^\circ$ for an irregular nonagon.

The correct way: only a regular nonagon has equal $140^\circ$ angles. An irregular one still sums to $1260^\circ$ but shares it unevenly. The same "assume it's regular" slip caused an early tiling error in Islamic geometric patterns, where nine-fold shapes had to be cut irregularly to close a repeating pattern that a perfectly regular nonagon cannot tile alone.

Conclusion

• A nonagon is a nine-sided polygon with 9 vertices, 9 angles, and 27 diagonals.

• Its interior angles always sum to $1260^\circ$; a regular nonagon has each angle equal to $140^\circ$.

• Each exterior angle of a regular nonagon is $40^\circ$, and interior plus exterior makes a straight $180^\circ$.

• Nonagons are classified as regular or irregular and, separately, as convex or concave.

To explore polygons and their angle rules with a teacher, try Bhanzu's geometry tutor or a middle school math tutor, or browse math classes online.

Practice These to Solidify Your Understanding

Work through the exercises below. Compute the interior sum for a nonagon; find each interior and exterior angle of the regular case; count its diagonals with the $\frac{n(n-3)}{2}$ formula; and work backwards from a $140^\circ$ angle to identify the shape. If interior and exterior angles keep swapping, reread the vertex diagram. To work through polygon problems live with a Bhanzu trainer, book a free demo class.

Read More

• Irregular Polygons — how a nonagon behaves when its sides and angles are unequal.

• Octagon — the eight-sided neighbour, with its own angle and diagonal counts.

• Heptagon — the seven-sided polygon, one step down from a nonagon.

• Area of Polygons — how to measure the space inside a nonagon and other polygons.

• Apothem — the inner radius you need for a regular nonagon's area.