# What is an Enneagon? - Definition & Formulas

Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson you will discover the nine-sided geometrical shapes called enneagons and learn formulas for perimeter, exterior and interior angles, radial length and apothem.

## Definition of Enneagon

The term enneagon refers to any nine sided polygon, but you may also have heard the term nonagon for this figure. The two words mean the same thing. Both ennea and nona are prefixes meaning 'nine' (ennea is Greek for 'nine', while nona comes from the Latin for 'ninth').

If all of the sides and angles of the enneagon are congruent, then it is called a regular enneagon. In this case, congruent just means equal in size and shape.

## Formulas That Apply to All Enneagons

But before we did into the formulas that only work on regular enneagons, let's first talk about the formulas that work for all enneagons.

• The perimeter of an enneagon is found by adding up all the side lengths. In fact, this how you find perimeter for any polygon, which is any closed, two-dimensional shape with three or more sides.
• The sum of the exterior angles is 360 degrees (also something that is true for any polygon).
• The sum of the interior angles is 1260 degrees. This result comes from the more general formula for the sum of interior angles of any n-sided polygon: (n - 2)(180). In the case of the enneagon, of course, n = 9.

Let's find the perimeter of the enneagon pictured below.

It's as simple as adding all the numbers - make sure you get all nine of them!

4.5 + 4 + 3.5 + 4 + 4 + 4.5 + 6 + 7.5 + 4 = 42.

The perimeter is 42.

## Regular Enneagons: Finding Perimeters and Angles

As mentioned before, all the sides and angles of a regular enneagon are congruent. In the picture above, every side length would be x. Now suppose you are given the length of one side and you want to know the perimeter. Easy! Because all the sides are the same length, just multiply by the length by the number of sides.

Perimeter = 9x

Let's find some angles now. An exterior angle (such as angle PBC) has a measure of 360 / 9 = 40 degrees. (Notice, all I did here was take the known sum of exterior angles, 360, and divided by the number of sides.) Similarly, an interior angle (such as angle ABC) has a measure of 1260 / 9 = 140 degrees.

## Radial Length and Apothem of a Regular Enneagon

The radial length of a polygon is the distance from the center to a vertex. In our enneagon figure, that's r. Every spoke of the wheel, like OA, OB, OC, etc. has the same radial length.

The apothem is the distance a from the center to the midpoint of a side - like segment OM in our figure.

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