por What is a hendecagon?
He hendecagonalso called undecagon, It is a closed flat geometric figure with 11 sides that belongs to the family of polygons.
These are named according to the number of sides they have and in the case of the hendecagon, its name derives from the Greek words «hendeka» and «gona»: eleven and vertex, respectively, according to the fact that the hendecagon has 11 vertices or points.
Regular polygons share a number of distinctive features, in particular the regular hendecagon, one whose sides all have the same length. Figure 1 shows a regular hendecagon and its most significant properties:
sidesa total of 11.
verticesthere are also 11 points that join two consecutive sides (in the previous figure they are the blue points and in figure 3 they are also named with capital letters).
Centerequidistant point from both the vertices and the sides.
diagonalslines that join a vertex with another non-consecutive vertex, in total of 44.
internal angles, those that are formed between two adjacent sides on the internal side of the hendecagon. If the hendecagon is regular, all the internal angles measure 147 3/11º.
external anglesare formed between one side and the extension of one of the consecutive sides.
Radiodistance from the center to a vertex.
central anglewhich in the case of the regular hendecagon measures 32 8/11º, whose sides are two adjacent segments and the vertex of the angle coincides with the center.
Apothemperpendicular segment that joins the center of a side with the center of the figure.
How to make a regular hendecagon?
To draw a regular hendecagon, whose sides measure the same, you need a ruler and a compass. One way to do the drawing is by following these steps:
1.- Draw a circle and two diameters of it, one vertical and one horizontal. The points of the circumference that determine these diameters are named A and B (horizontal diameter) and C and D (vertical diameter).
2.- Open the compass with the measurement of the radius of the circumference, rest the point at point D of the diameter CD and draw a first arc that intersects the circumference at point E.
3.- With this same measurement, support the point of the compass at point A and draw a second arc that intersects the circumference at point F and at the same time passes through its center.
4.- Open the compass with the point resting on E and up to point F, tracing a third arc that cuts the vertical diameter at point G.
5.- Now, open the compass between points F and G. This will be the measure of the side of the hendecagon. The point of the compass is supported on F and a fourth arc is drawn that cuts the circumference at point H, the side FH already belongs to the hendecagon and is drawn joining the points with the help of the ruler.
6.- The point of the compass rests successively on point H and arcs are carefully traced, joining the points thus determined by means of segments, until completing the eleven sides of the polygon.
Examples of hendecagons
There are several kinds of hendecagons, depending on the measure of their sides and their internal angles, here are some examples:
regular and irregular hendecagons
–Regularif all the sides and interior angles measure the same.
–Irregularwhen their sides have different measures.
The figure below shows a regular hendecagon on the inner contour of a US dollar with the figure of Susan B. Anthony (1820-1906), a women’s rights activist born in Massachusetts, United States. The hendecagon has also been used as part of the design of other coins throughout the world.
convex and concave hendecagons
The hendecagons differ according to their internal angles, for example they can be:
–convexif the internal angles are less than 180º.
–concavewhen there are internal angles greater than 180º.
The hendecagon that decorates the Susan B. Anthony dollar is convex, since the measure of any of its internal angles is less than 180º. Its value is calculated through a formula that depends on the number of sides of the figure (see the next section).
Formulas for the hendecagon
Formula for internal angles
To determine the measure I in degrees, of the interior angles of any regular polygon of no sides, the following formula is used:
Substituting the number of sides of the hendecagon n = 11 in the formula we obtain:
And since the internal angles of the regular hendecagon are less than 180º, this figure is a convex polygon.
The sum S of the internal angles of a regular polygon is found by this formula, valid for n integer and greater than 2:
S = (n−2) x 180º
Substituting n=11 results:
S = (11 – 2) x 180º = 1620º
external angles
To know the measure of internal angles, it is applied that the sum of an internal angle and an external angle is equal to 180º:
180º – 147 3/11º = 32 8/11º.
Perimeter
The perimeter is the sum of the sides of the hendecagon, whether regular or not. In the case of the regular hendecagon, if ℓ is the length of one of the sides, the perimeter is found by multiplying by nothe number of sides.
Therefore the perimeter P of the regular hendecagon is:
P = 11 ℓ
Area
Knowing the side, the area can be calculated with the formula:
which is approximately:
A = 9.3656∙ℓ2
Another way to find the area, as long as the hendecagon is regular, consists of dividing it into triangles with base equal to side ℓ and height equal to the length of apothem LA.
The area of each triangle is calculated by:
Area = base x height /2
Which depending on the apothem is also written as:
Area = ℓ. THE /2
And the total area of the hendecagon is found by multiplying the area of the triangle by 11:
A=11ℓ. THE /2
In terms of the perimeter, the area of the regular hendecagon is:
A = LA P. /2
diagonals
The number of diagonals is calculated by setting n = 11 in the following formula:
D = [11 x (11-3)]/2 = 44 diagonals.
solved exercise
Calculate perimeter and area of a regular hendecagon with sides 20 cm.
Solution
The perimeter is:
P = 11 ℓ =11×20cm = 220cm.
And its area is:
A = 9.3656∙ℓ2=9.3656×(20 cm)2=3746.2 cm2
References
Alexander, D. 2013. Geometry. 5th. Edition. Cengage Learning.
Drawing teacher. Hendecagon inscribed in a circle (11-sided polygon). Recovered from: youtube.com.
Problems and equations. Calculator of the area and perimeter of the regular hendecagon. Recovered from: problemasyecuaciones.com.
Sangaku Maths. Elements of a polygon and their classification. Recovered from: sangakoo.com.
Automatic geometry problem solver. The hendecagon. Retrieved from: scuolaelettrica.it.
Wolfram Math World. Hendecagon. Retrieved from: mathworld.wolfram.com.