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NY Regents Exam - Geometry: Help and Review11 chapters | 135 lessons | 8 flashcard sets

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.

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.

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.

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 = 9*x*

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.

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.

In more advanced courses, such as trigonometry, you may be asked to find the radial length and apothem of a regular polygon. This is quite easy if you know your trig functions! Notice that triangle *OMC* is a right triangle. We know the measure of angle *OCM* is 70 degrees (because it is exactly half of an interior angle of 140 degrees). Lastly, the length of segment *CM* is *x*/2 (because *M* is the midpoint of *CD*). Putting all together with the appropriate trig functions, we find:

cos(70) = (*x*/2) / *r*

tan(70) = *a* / (*x*/2)

Solving for the unknowns (*r* and *a*, respectively), we discover the formulas for the enneagon:

- Radial length:

- Apothem:

The formulas are super easy to use, even if you don't know trig! For example, suppose your enneagon is regular and has side length 20 cm. Find the radial length and apothem.

Radial length = (1.4619)(20) = 29.238 cm.

Apothem = (1.3738)(20) = 27.476 cm.

An enneagon, also known as a nonagon, is any nine-sided polygon. If all of the sides and angles of the enneagon are congruent, then the enneagon is called regular. A regular enneagon has exterior angles that measure 40 degrees and interior angles of 140 degrees. These are the formulas for radial length and apothem of a regular enneagon of side length *x*:

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NY Regents Exam - Geometry: Help and Review11 chapters | 135 lessons | 8 flashcard sets

- The Parallel Postulate: Definition & Examples 4:25
- Angles Formed by a Transversal 7:40
- Parallel Lines: How to Prove Lines Are Parallel 6:55
- Using Converse Statements to Prove Lines Are Parallel 6:46
- What Are Polygons? - Definition and Examples 4:25
- Measuring the Area of Regular Polygons: Formula & Examples 4:15
- Measuring the Angles of Triangles: 180 Degrees 5:14
- How to Measure the Angles of a Polygon & Find the Sum 6:00
- What is an Enneagon? - Definition & Formulas
- Hexagonal Prism: Properties, Formula & Examples 6:49
- Go to NY Regents - Parallel Lines and Polygons: Help and Review

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