Nonagon — Definition, Regular & Irregular Examples

Written by
Malcolm McKinsey
Fact-checked by
Paul Mazzola

What is a nonagon?

nonagon or 9-gon is a nine-sided polygon with nine interior angles that sum to 1260°. A nonagon is also called an enneagon. In many ways, enneagon is a more accurate term, since it uses Greek roots for both its prefix (ennea-, meaning nine) and suffix (-gon, angled).

Nonagons are rarely used in most geometry texts, but it is an intriguing shape, whether you examine a regular nonagon or an irregular nonagon.

Nonagon shape

To be a nonagon or enneagon, the shape must have all these properties:

• It must be two dimensional (planar)

• It must use only nine line segments

• It must close in an area

The nine line segments are the nonagons' sides, and they connect to form the nonagons' angles.

Nonagon sides

A nonagon has nine straight sides of equal length. If the sides are all the same length, then you have a regular nonagon. If the sides vary in length, then you have an irregular nonagon.

If the sides are curved, it is not a nonagon.

Nonagon angles

Every nonagon has nine interior angles that sum to 1260°, and the nine exterior angles of a nonagon sum to 360°.

Every simple nonagon (not complex) can have it's interior divided into seven triangles.

You can find the sum of interior angles by multiplying 7 × 180°, which is how you get 1260°.

Nonagon diagonals

Because every diagonal has 9 interior angles that connect, a nonagon has 27 diagonals.

Regular nonagon

A regular nonagon shape has 9 equal sides and 9 congruent angles. In a regular nonagon, each interior angle measures 140°.

A regular nonagon is a subtle and beautiful design:

It is very nearly a circle, but not quite. The interior angles are each 140°. You can verify this because 1260° ÷ 9 (angles) = 140°.

The central angle is 40° (which you can calculate by dividing 360° by 9). This means you can construct a regular nonagon with 7 isosceles triangles with two angles of 70° and one angle of 40°.

Irregular nonagon

Irregular nonagons can have sides of different lengths and interior angles of varying sizes. Yet, even odd shapes like these are nonagons:

A polygon with at least one interior angle greater than 180° is a concave polygon and always created an irregular polygon.

Irregular nonagons and regular nonagons both share these same facts:

• Its nine interior angles sum to 1260°

• It has nine sides, even if those sides cross one another.

• Its nine exterior angles sum to 3660°

Nonagon shape in real life

Nonagon examples are rare in both the natural and human-made worlds. Two exceptions:

1. The U.S. Steel Tower in Pittsburgh, Pennsylvania, is an irregular nonagon.

2. All houses of worship in the Bahá’í faith are regular nonagons.

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