- Properties of the eneagon
- Regular enegon
- Area of an enegon known the side and the apothem
- Area of a known enegon the side
- Perimeter of regular enegon known its side
- Perimeter of the enegon known its radius
- How to make a regular enegon
- Examples
- Example 1
- Example 2
- References
An enegon 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 eneagon are unequal to each other, then you have an irregular eneagon. If, on the other hand, all nine sides and nine angles of the eneagon are equal, then it is a regular eneagon.
Figure 1. Regular eneagon and irregular eneagon. (Own elaboration)
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 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.
The measure of the internal angles of an enegon is then 1260º / 9 = 140º.
Figure 2. Apothem, radius, sides, angles, and vertices of a regular eneagon. (Own elaboration)
To derive the formula for the area of a regular enegon with side d, it is convenient to make some auxiliary constructions, such as those shown in figure 2.
The center O is found by tracing the bisectors of two adjacent sides. The center O equidistant from the vertices.
A radius of length r is the segment from the center O to a vertex of the enegon. Figure 2 shows the radii OD and 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 a.
Area of an enegon known the side and the apothem
We consider the triangle ODE in figure 2. The area of this triangle is the product of its base DE and the height OJ divided by 2:
ODE area = (DE * OJ) / 2 = (d * a) / 2
Since there are 9 triangles of equal area in the enegon, it is concluded that the area of the same is:
Enegon 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 in order to apply the formula in the previous section.
We consider the right triangle OJE in J (see figure 2). If the tangent trigonometric ratio is applied, we obtain:
tan (∡ OEJ) = OJ / EJ.
The angle ∡OEJ = 140º / 2 = 70º, since EO is the bisector of the internal angle of the enegon.
On the other hand, OJ is the apothem of length a.
Then, since J is the 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:
Area of the enegon = (9/2) (d * a) = (9/2) (d * (d / 2) tan (70º))
Finally, we find the formula that allows obtaining the area of the regular enegon if only the length d of its sides is known:
Area of the enegon = (9/4) d 2 tan (70º) = 6.1818 d 2
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, that is:
Perimeter = 9 d
Perimeter of the enegon known its radius
Considering the right triangle OJE in J (see figure 2), the trigonometric cosine ratio is applied:
cos (∡ OEJ) = EJ / 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 enegon
1- To build a regular eneagon, with a ruler and a compass, start from the circumference c that circumscribes the eneagon. (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 at the intercept B and the opening equal to the radius BO, an arc is drawn that intercepts the original circumference at a point C.
Figure 3. Steps to build a regular enegon. (Own elaboration)
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 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-out examples.
Example 1
We want to build a regular enegon whose sides measure 2 cm. What radius must have the circumference that circumscribes it, so that by applying the construction described previously the desired result is obtained?
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 gives a radius r of 2.92 cm.
Example 2
What is the area of a regular enegon with a side 2 cm?
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:
Area of the enegon = (9/4) d 2 tan (70º) = 6.1818 d 2
Substituting d for its value of 2 cm in the previous formula, we obtain:
Eneagon area = 24.72 cm
References
- CEA (2003). Geometry elements: with exercises and compass geometry. University of Medellin.
- Campos, F., Cerecedo, FJ (2014). Mathematics 2. Grupo Editorial Patria.
- Freed, K. (2007). Discover Polygons. Benchmark Education Company.
- Hendrik, V. (2013). Generalized Polygons. Birkhäuser.
- IGER. (sf). Mathematics First Semester Tacaná. IGER.
- Jr. geometry. (2014). Polygons. Lulu Press, Inc.
- Miller, Heeren, & Hornsby. (2006). Mathematics: Reasoning And Applications (Tenth Edition). Pearson Education.
- Patiño, M. (2006). Mathematics 5. Editorial Progreso.