Eneagon properties, how to make an eneagon, examples

A eneagon is a polygon with nine sides and nine vertices, which may or may not be regular. The name eneágono comes from the Greek and is made up of the Greek words ennea (nine and gonon (angle).

An alternative name for the nine-sided polygon is nonagon, which comes from the Latin word nonus (nine and gonon (vertex). On the other hand, if the sides or angles of the enegon are unequal to each other, then we have a irregular eneagon. If, on the other hand, the nine sides and the nine angles of the enegon are equal, then it is a regular enegon.

Article index

• 1 Properties of the enegon
  • 1.1 Regular enegon
  • 1.2 Area of ​​an enegon known the side and the apothem
  • 1.3 Area of ​​a known enegon the side
  • 1.4 Perimeter of the regular enegon known its side
  • 1.5 Perimeter of the enegon known its radius
• 2 How to make a regular eneagon
• 3 Examples
  • 3.1 Example 1
  • 3.2 Example 2
• 4 References

Properties of the eneagon

For a polygon with n sides the sum of its interior angles is:

(n - 2) * 180º

In the enegon it would be n = 9, so the sum of its internal angles is:

Sa = (9 - 2) * 180º = 7 * 180º = 1260º

In any polygon, the number of diagonals is:

D = n (n - 3) / 2 and in the case of the enegon, since n = 9, we then have that D = 27.

Regular enegon

In the regular eneagon or nonagon there are nine (9) internal angles of equal measure, therefore each angle measures one-ninth of the total sum of the internal angles.

We then have that the measure of the internal angles of an enegon is 1260º / 9 = 140º.

To derive the formula for the area of ​​a regular enegon with side d it is convenient to make some auxiliary constructions, like those shown in figure 2.

The center is located OR tracing the bisectors of two adjacent sides. The center OR equidistant from vertices.

A radius of length r is the segment that goes from the center OR to a vertex of the enegon. The radii are shown in figure 2. OD Y OE of length r.

The apothem is the segment that goes from the center to the midpoint of one side of the enegon. For example OJ is an apothem whose length is to.

Area of ​​an enegon known the side and the apothem

We consider the triangle ODE of figure 2. The area of ​​this triangle is the product of its base FROM for the height OJ divided by 2:

Area ODE = (DE * OJ) / 2 = (d * a) / 2

Since there are 9 triangles of equal area in the enegon, it is concluded then that the area of ​​the same is:

Eneagon area = (9/2) (d * a)

Area of ​​a known enegon the side

If only the length d of the sides of the enegon is known, then it is necessary to find the length of the apothem to be able to apply the formula from the previous section.

We consider the triangle EYE rectangle in J (see figure 2). If the tangent trigonometric ratio is applied, we obtain:

so(∡OEJ) = OJ / Ex.

The angle ∡OEJ = 140º / 2 = 70º, to be EO bisector of the internal angle of the enegon.

On the other hand, OJ is the apothem of length to.

Then as J is midpoint of ED it follows that EJ = d / 2.

Substituting the previous values ​​in the tangent relation we have:

tan (70º) = a / (d / 2).

Now we clear the length of the apothem:

a = (d / 2) tan (70º).

The previous result is substituted in the area formula to obtain:

Eneagon area = (9/2) (d * a) = (9/2)( d * (d / 2) tan (70º))

Finally, there is the formula that allows obtaining the area of ​​the regular enegon if only the length is known d from its sides:

Eneagon area = (9/4) d^two tan (70º) = 6.1818 d^two

Perimeter of regular enegon known its side

The perimeter of a polygon is the sum of its sides. In the case of the enegon, as each and every one of the sides measures a length d, its perimeter will be the sum of nine times d, namely:

Perimeter = 9 d

Perimeter of the enegon known its radius

Considering the triangle EYE rectangle in J (see figure 2), the trigonometric cosine ratio is applied:

cos (∡OEJ) = Ex / OE = (d / 2) / r

Where is it obtained from:

d = 2r cos (70º)

Substituting this result, we obtain the formula for the perimeter as a function of the radius of the enegon:

Perimeter = 9 d = 18 r cos (70º) = 6.1564 r

How to make a regular eneagon

1- To build a regular eneagon, with a ruler and a compass, start from the circumference c that circumscribes the enegon. (see figure 3)

2- Two perpendicular lines are drawn through the center O of the circumference. Then the intersections A and B of one of the lines are marked with the circumference.

3- With the compass, centering on the intercept B and opening equal to the radius BO, an arc is drawn that intercepts the original circumference at a point C.

4- The previous step is repeated but making a center at A and radius AO, an arc is drawn that intercepts the circumference c at point E.

5- With opening AC and center in A, an arc of circumference is drawn. Similarly with opening BE and center B another arc is drawn. The intersection of these two arcs is marked as point G.

6- Centering at G and opening GA, an arc is drawn that intercepts the secondary axis (horizontal in this case) at point H. The intersection of the secondary axis with the original circumference c is marked as I.

7- The length of the segment IH is equal to the length d of the side of the enegon.

8- With compass opening IH = d, the arcs of center A radius AJ, center J radius AK, center K radius KL and center L radius LP are drawn successively.

9- Similarly, starting from A and from the right side, arcs of radius IH = d are drawn that mark the points M, N, C and Q on the original circumference c.

10- Finally the segments AJ, JK, KL, LP, AM, MN, NC, CQ and finally PB are drawn.

It should be noted that the construction method is not entirely exact, since it can be verified that the last side PB is 0.7% longer than the other sides. To date, there is no known method of construction with a ruler and compass that is 100% accurate..

Examples

Here are some worked examples.

Example 1

You want to build a regular enegon whose sides measure 2 cm. What radius must have the circumference that circumscribes it, so that when applying the construction described previously, the desired result is obtained?

Solution:

In a previous section, the formula that relates the radius r of the circumscribed circle with the side d of a regular enegon was deduced:

d = 2r cos (70º)

Solving for r from the previous expression we have:

r = d / (2 cos (70º)) = 1.4619 * d

Substituting the value d = 2 cm in the previous formula, a radius r of 2.92 cm is obtained.

Example 2

What is the area of ​​a regular enegon with a side 2 cm?

Solution:

To answer this question, we must refer to the formula, previously shown, which allows us to find the area of ​​a known enegon by the length d of its side:

Eneagon area = (9/4) d^two tan (70º) = 6.1818 d^two

Substituting d for its value of 2 cm in the previous formula, we obtain:

Eneagon area = 24.72 cm

References

1. C. E. A. (2003). Elements of geometry: with exercises and geometry of the compass. University of Medellin.
2. Campos, F., Cerecedo, F. J. (2014). Mathematics 2. Grupo Editorial Patria.
3. Freed, K. (2007). Discover Polygons. Benchmark Education Company.
4. Hendrik, V. (2013). Generalized Polygons. Birkhäuser.
5. IGER. (s.f.). Mathematics First Semester Tacaná. IGER.
6. Jr. geometry. (2014). Polygons. Lulu Press, Inc.
7. Miller, Heeren, & Hornsby. (2006). Mathematics: Reasoning And Applications (Tenth Edition). Pearson Education.
8. Patiño, M. (2006). Mathematics 5. Editorial Progreso.