Regular Nonagon — Definition, Formula & Examples

A regular nonagon is a nine-sided polygon where every side has the same length and every interior angle has the same measure.

A regular nonagon is an equilateral and equiangular polygon with exactly nine vertices, nine sides of equal length, and nine interior angles each measuring 140°.

Key Formula

\text{Each interior angle} = \frac{(n - 2) \times 180°}{n}

Where:

$n$ = Number of sides (for a nonagon, n = 9)

How It Works

Since a nonagon has 9 sides, you can find its total interior angle sum using the polygon angle formula:

(n - 2) \times 180°

, which gives

(9 - 2) \times 180° = 1{,}260°

. Because the nonagon is regular, each interior angle equals

1{,}260° \div 9 = 140°

. Each exterior angle measures

180° - 140° = 40°

, and all nine exterior angles sum to

360°

as with any convex polygon.

Worked Example

Problem: A regular nonagon has a side length of 6 cm. How many diagonals does it have, and what is the measure of each interior angle?

Find the number of diagonals: Use the diagonal formula for any polygon:

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

Find each interior angle: Apply the interior angle formula with n = 9:

\frac{(9 - 2) \times 180°}{9} = \frac{1{,}260°}{9} = 140°

Answer: A regular nonagon has 27 diagonals, and each interior angle measures 140°.

Why It Matters

Nonagons appear in architecture and tiling patterns. Practicing with 9-sided polygons strengthens your ability to apply the general polygon angle and diagonal formulas to any number of sides.

Common Mistakes

Mistake: Confusing a nonagon (9 sides) with a decagon (10 sides) or an octagon (8 sides).

Correction: Remember the prefix: "nona-" means nine, just like "November" was originally the ninth month in the Roman calendar.