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Nonagon — Definition, Properties & Examples

Nonagon

A polygon with nine sides.

 

Irregular nonagon (9-sided polygon) with unequal sides and angles, shown as an example of a non-regular nonagon.

Nonagon

A regular polygon with 9 sides (nonagon) showing equal sides and angles, no labels.

Regular Nonagon

Key Formula

S=(n−2)×180∘=(9−2)×180∘=1260∘S = (n - 2) \times 180^\circ = (9 - 2) \times 180^\circ = 1260^\circ
Where:
  • SS = Sum of interior angles of the nonagon
  • nn = Number of sides (9 for a nonagon)

Worked Example

Problem: Find the measure of each interior angle and each exterior angle of a regular nonagon.
Step 1: Use the interior angle sum formula for any polygon with n sides.
S=(n−2)×180∘S = (n - 2) \times 180^\circ
Step 2: Substitute n = 9 to find the total sum of interior angles.
S=(9−2)×180∘=7×180∘=1260∘S = (9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ
Step 3: Since a regular nonagon has all angles equal, divide the sum by 9 to find each interior angle.
Each interior angle=1260∘9=140∘\text{Each interior angle} = \frac{1260^\circ}{9} = 140^\circ
Step 4: Each exterior angle and its adjacent interior angle are supplementary (they add to 180°).
Each exterior angle=180∘−140∘=40∘\text{Each exterior angle} = 180^\circ - 140^\circ = 40^\circ
Answer: Each interior angle of a regular nonagon measures 140°, and each exterior angle measures 40°.

Another Example

Problem: A regular nonagon has a side length of 6 cm. Find the number of diagonals it contains.
Step 1: Use the diagonal formula for any polygon: the number of diagonals depends only on n, the number of vertices.
d=n(n−3)2d = \frac{n(n - 3)}{2}
Step 2: Substitute n = 9.
d=9(9−3)2=9×62=542=27d = \frac{9(9 - 3)}{2} = \frac{9 \times 6}{2} = \frac{54}{2} = 27
Answer: A nonagon has 27 diagonals.

Frequently Asked Questions

How many sides does a nonagon have?
A nonagon has exactly 9 sides. The prefix "nona-" comes from the Latin word for nine. It also has 9 vertices (corners) and 9 interior angles.
What is the sum of angles in a nonagon?
The sum of the interior angles of any nonagon is 1260°. You can find this using the formula (n − 2) × 180° with n = 9. For a regular nonagon, each individual interior angle is 1260° ÷ 9 = 140°.

Nonagon (9 sides) vs. Decagon (10 sides)

A nonagon has 9 sides with an interior angle sum of 1260°, while a decagon has 10 sides with an interior angle sum of 1440°. In regular forms, each interior angle of a nonagon is 140° and each interior angle of a decagon is 144°. Both are named using Latin-derived prefixes.

Why It Matters

Nonagons appear in architecture, tiling patterns, and coin design — for example, some countries use nonagonal coins that are easy to identify by touch. Understanding nonagons reinforces the general polygon angle formulas that apply to any shape. Studying polygons with more sides also builds toward the idea that regular polygons approach a circle as the number of sides increases.

Common Mistakes

Mistake: Confusing a nonagon (9 sides) with an octagon (8 sides) or a decagon (10 sides).
Correction: Remember the prefix: "nona-" means nine (like "nonagenarian" for a person in their nineties). An octagon has 8 sides and a decagon has 10.
Mistake: Using n = 9 directly in the angle sum formula as 9 × 180° instead of (9 − 2) × 180°.
Correction: The formula requires you to subtract 2 from the number of sides first. A nonagon can be divided into 7 triangles (not 9), giving a sum of 7 × 180° = 1260°.

Related Terms