Regular polygon . In Mathematics and Geometry they are the convex polygonal figures that have all their sides and interior angles equal.
The n- sided polygon is also called the regular n-agon .
Another characteristic of capital importance is the fact that also its interior and exterior angles are congruent, which greatly facilitates the calculation of the perimeter, area; integral parts, aspects that have led throughout the development of geometry and used in the calculation of other more complex surfaces; in such circumstance they are seen as more particular cases, the n-agons. Furthermore, their apothems are the same, and the radii of the inscribed or circumscribed circles can be calculated, even trigonometrically.
The regular polygon with fewer sides is the equilateral triangle , followed by squares , regular pentagons , etc.
It should not be confused with the concept of an equilateral polygon (polygons whose edges are all equal in length), as there are some that are convex but not regular, as is the case of the rhombus and others that have concavities.
Summary
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- 1 Definitions
- 2 Examples
- 3 Properties
- 4 Calculation
- 1 Component elements
- 2 Perimeter
- 3 Area
- 3.1 Areas of the circumscribed and inscribed circles
- 4 Diagonals
- 5 Graphical representation
- 6 Importance
- 7 Notes
- 8 Sources
Definitions
A polygon is said to be regular if and only if its sides and interior angles are equal to each other.
Description of the elements of a regular polygon |
· l : Length of the edges of the polygon.
· r : Radius of the circumscribed circumference . · n . Number of sides of the polygon. · : Interior angle of the polygon · h : Radius of the inscribed circumference or apothem . · : Central angle of talque |
Examples
Below is a table with some regular polygons and some of their general values.
N | Name | Figure | Central angles | Interior angles | Perimeter |
3 | equilateral triangle | 120 ^{o} | 60 ^{o} | 3l | |
4 | square | 90 ^{o} | 90 ^{o} | 4l | |
5 | pentagon | 72 ^{or} | 108 ^{or} | 5l | |
6 | hexagon or hexagon | center | 60px | 60 ^{o} | 120 ^{o} | 6l |
7 | heptagon | 51,428571 ^{or} | 128.571428 ^{or} | 7l | |
8 | octagon | 45 ^{o} | 135 ^{o} | 8l | |
9 | nonagon | 40 ^{o} | 140 ^{o} | 9l | |
10 | decagon | 36 ^{o} | 144 ^{or} | 10l | |
eleven | hendecagon or undecagon | 32.72 ^{or} | 147.27 ^{or} | 11l | |
12 | dodecagon | 30 ^{o} | 150 ^{o} | 12l | |
fifty | quincuagon | 7.2 ^{or} | 172.8 ^{or} | 50l |
It can be seen how by increasing the number of sides the polygon tends to resemble a circle.
Properties
- Every regular polygon is equilateraland equiangular .
- The rhombusand the non-square rectangle are not regular polygons , even though the first is equilateral and the second is equiangular.
- If we divide a circle into nequal parts, where n is not less than 3, and we join the division points consecutively by segments, we obtain an inscribed regular n -agon .
- If we divide a circumference into nequal parts, where n is not less than 3, and draw consecutively by the tangent straight division points (each one cuts two of them) that intersect, we obtain a circumscribed regular n -agon . ^{[one]}
Calculation
Regular polygons have a series of properties that make it extremely easy to calculate their constituent elements in addition to the perimeter and area.
The variables that appear have been referred from the definitions section
Component elements
Defining regular n-agons allows close relationships to exist between their geometric components so that if some of them are known, the rest can be obtained.
- Algebraic ratio of r, h and l : 4r ^{2} = 4h ^{2} + l ^{2} .
- Width of interior angles:
- Amplitude of the central angles: .
- Amplitude of the exterior angles: .
- .
- .
- .
- .
Perimeter
Obviously the most trivial way to obtain the perimeter of a regular n-gon is knowing the length l of one of its sides, in which case:
- L _{n}= nl
Depending on the radius of the circumscribed circle r, the formula adjusts to:
Area
To determine the area of a regular n-gon, the most practical is to subdivide it into triangles, drawing the radii r of the circumscribed circumference towards each of the vertices, which results in n equal isosceles triangles (the radii are equal and by definition, the edges are congruent) to each other, so that if the area A _{T} of one of these triangles is known, the general surface can be calculated by:
- A _{n}= nA _{T}
Meanwhile, A _{T} can be determined by the general formula for the area of triangles:
Summarizing:
In another situation, if we had the radius of the circumscribed circle r , the total surface expression is:
Or this one that depends on the apothem and the central angle:
Circumscribed and inscribed circumference areas
If the length is known l of the n-gon can regulate calculated values to corresponding surfaces circumscribed and inscribed circles.
Kind | Area | Perimeter |
Registered circumference | ||
Circumference circumscribed |
Diagonals
Being n the number of sides of the polygon; n not less than 3
Number of diagonals
N _{d} = n (n-3) / 2
Number of intersections of diagonals
N _{Id} = n (n-1) (n-2) (n-3) / 24 ^{[2]}
Graphic representation
The graphical representation of regular n-agons has been of particular interest in geometry, architecture , design , technical drawing , plastic arts , religion , generation of both 2D and three-dimensional computer graphics and many others, therefore over time it They have developed methods of all kinds for obtaining it.
Of great interest from the great cultures of antiquity was obtaining them with simple instruments as a ruler and compass , achieving with some speed the construction of the regular polygons of 3, 4, 5, 6, 8 and 15 sides with the high geometric development achieved in Greece of Euclides , although similar forms have in classical texts of ancient India , China and Egypt , besides the cuneiform tablets of Mesopotamia .
However, there were figures that resisted construction with a ruler and compass, such as the heptagon, which more than 2,000 years later would prove the inability to obtain it only with a ruler and compass.
The regular polygons that can be obtained with these instruments are known as buildable polygons . The main idea is to divide a circle into n equal sections, a phenomenon called cyclotomy from the Greek cyclos (circle) and tome (divide, divisible).
In 1796 Gauss demonstrates that it is also possible to build with a ruler and compass the heptadecagon (17-sided regular n-gon) and five years later he had established a sufficient condition for a regular polygon to be constructible by establishing an association between the number of sides and the currently called prime numbers of Fermat .
Obviously, analytic geometry presented simpler algebraic solutions. The idea is the same as an n- part cyclotomy that assuming that the circle of radius r has its origin at the coordinate center (0; 0) the vertices of the regular n-gon are located at the points:
- for i = 0..n.
If you want a more general version that includes rotating at an angle .
- for i = 0..n.
Moving to another coordinate (x, y) of the center only requires adding these values to all the coordinates of the vertices:
- for i = 0..n.
Importance
The knowledge and properties of these geometric figures transcends the field of mathematics where they have constituted an abstract and aesthetic fascination for the simplicity and harmonic regularity of their forms, to reach other areas of knowledge and practices of Humanity.
It is easy to see that a regular polygon with more sides has more resembles a circle, and that its infinity in fact constitutes the circle. This result helped in antiquity to obtain approximations of the number pi by means of the exhaustive method where the measurement of the lengths of regular unit n-agos was taken.
They are also useful for calculating the surface and volume of more complex figures by inscribing regular n-ags of known formulas in them. One of the graphical visual expression methods is based on the cyclotomy of regular polygons.
In nature there are materials, particles , molecules , waves that have the appearance of these figures or their behavior can be modeled using the formulas that govern the n-agon. Examples of this are the hexagonal shapes of ice crystals in snowflakes or the graphite and benzene molecules .
Drawing, plastic arts and design make use of these for various aesthetic, expressive or conceptual purposes, as some of these figures are part of the general culture or of nations, ethnic groups or religions, symbolically associating their forms with ideas, concepts, phenomena , abstractions, movements, trends, fashions, etc.
Computer imaging (GIC, CGI in English ), video games , AVID editing and 3D graphics almost always have at the most elemental polygons or polyhedra whose very comfortable properties favor the speed of calculation for an inexpensive obtaining in terms of power calculation of each table, so that the always desirable property is met in real time, what you see is what you will get (WYSIWYG, for its English equivalent).