# Nonagon

## Definition

A nonagon is a polygon with nine sides and nine angles. The term ‘nonagon’ is a hybrid of the Latin word ‘nonus’ meaning nine and the Greek word ‘gon’ meaning sides. It is also known as ‘enneagon’, derived from the Greek word ‘enneagonon’, also meaning nine. Although ‘enneagon’ is the more appropriate name for this shape, the name ‘nongon’ is more used for simplicity and convenience.

The below-given properties and formulas to find the perimeter and area only apply to the regular nonagon.

## Properties

1. Has 9 sides of equal length; in nonagon ABCDEFGHI, AB = BC = CD = DE = EF = FG = GH = HI = IA
2. Has 9 interior angles each measuring 140°; so ∠ABC = ∠BCD = ∠CDE = ∠DEF = ∠EFG =∠FGH = ∠GHI = ∠HIA = ∠IAB
3. The sum of all nine interior angles is 1260°; so ∠ABC + ∠BCD + ∠CDE + ∠DEF + ∠EFG + ∠FGH + ∠GHI + ∠HIA + ∠IAB = 1260°
4. Has 9 exterior angles, each measuring 40°
5. Has 27 diagonals; example AC, BD, CE, DF, EG, FH, and GI

## Formulas

### Perimeter

The formula for finding the perimeter of a nonagon is given below:

Find the perimeter of a regular nonagon with each side measuring 11 cm.

Solution:

As we know,
Perimeter (P) = 9a, here a = 11 cm
= 9 x 11 cm
= 99 cm

### Area

The formula for finding the area of a nonagon is given below:

The above equation is approximately equal to

A = 6.182 a2 square units, here cot π/9 = cot 20 = 2.74

Find the area of a regular nonagon having each side measuring 9 cm.

Solution:

As we know,
Area (A) = 7/4 (a2cot π/7)
= 6.182 a2, here a = 9 cm
= 3.634 (9)2
= 3.634 (81)
= 294.35 cm2

### Angles

#### Interior Angle

The angle formed inside the nonagon at its corners when the line segments join in an end to end fashion.

Sum of Interior Angles

The total measure of all the interior angles combined in the nonagon. The formula is given below:

Sum of the interior angles = (n-2) x 180°, here n = number of sides

In nonagon ABCDEFGHI, n = 9

Thus,

Sum of the interior angles = (9 -2) x 180°

= 1260°

One Interior Angle

The measure of one interior angle can be obtained by dividing the sum of the interior angles by the number of sides in a nonagon. The formula is given below:

One interior angle = (n-2) x 180°/n, here n = number of sides

In nonagon ABCDEFGHI, n = 9

Thus,

Sum of the interior angles = (9-2) x 180°/9

= 140°

#### Exterior Angle

The angle formed by any side of the nonagon and the extension of its adjacent side. The formula is given below:

Exterior angle = 360°/n, here n = number of sides

In nonagon ABCDEFGHI, n = 9

Thus,

Exterior angle = 360°/9

= 40°

## Types

Depending on the sides, angles, and vertices, nonagon shapes are classified into the following types:

1. Regular Nonagon: Has nine sides of equal length and nine interior angles, each measuring 140°, and exterior angles of 40° each. It has nine lines of symmetry and rotational equilibrium of order nine. All regular nonagon is convex.
2. Irregular Nonagon: Does not have all sides equal or all interior angles equal, but the sum of all nine interior angles is equal to 1260°. An irregular nonagon can be both convex and concave.
3. Convex Nonagon: Have all nine vertices pointing outwards. No interior angle of a convex nonagon measure more than 180°, and all the diagonals lie inside the closed figure. A convex nonagon can be both regular and irregular.
4. Concave Nonagon: Have at least one vertex pointing inwards with an interior angle greater than 180°. At least one diagonal lies outside the closed figure. Thus all concave nonagons are irregular.

## Real Life Examples

• The shape of Bahá’í House of Worship in Sydney, Panama, Wilmette, Kampala, and in many other places all over the world.
• The shape of the US Steel Building in Pittsburgh, Pennsylvania, is an irregular nonagon.
• Octagonal-shaped objects such as coins, rings, pipes, cans, water bottles, and the bottom of a drinking glass.