The regular polygons are those that have all their sides and their internal angles equal. In the following figure there is a set of different polygons, which are plane figures limited by a closed curve and only those that are highlighted meet the conditions to be regular.
For example, the equilateral triangle is a regular polygon, since its three sides measure the same, as well as its internal angles, which are worth 60º each..
The square is a quadrilateral with four sides of equal measure and whose internal angles are 90º. It is followed by the regular pentagon, with five sides of equal size and five internal angles of 108º each..
When a polygon is regular, this word is added to its special name, so we have the regular hexagon, the regular heptagon and so on.
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The most important properties of regular polygons can be summarized as follows:
-The sides measure the same, therefore they are equilateral.
-They are equiangular, since all its internal angles have the same measure.
-They can always be inscribed in a circumference, which means that they fit perfectly within one, which is called circumscribed circumference.
-For a regular polygon with n sides, the measure of an interior angle α is:
α = [180 (n-2)] / n
-You can draw n (n-3) / 2 diagonals from the vertices of a polygon, whether it is regular or not.
-The sum of the exterior angles is equal to 360º.
Next we present the main elements of a regular polygon, visualized in the figure below.
Common point that two consecutive sides have, denoted as V in the figure.
It is the segment that joins two consecutive vertices of the polygon and is denoted as ℓ or L.
Segment that joins two non-consecutive vertices of the polygon, in the figure it is denoted as d.
It is the common center of the inscribed circle and the circumscribed circle, denoted by the letter O. It can also be seen as the only point equidistant from both the vertices and the midpoints of each side..
It's the radio r of the circumscribed circle and coincides with the distance between O and a vertex.
It is called apothem to the radius of the circumference inscribed in the polygon, represented in the figure with a letter to. The apothem is perpendicular to one side and joins it with the center O (red segment in figure 3).
Knowing the radius r and the length of the side, the apothem is calculated by:
Since, in effect, the apothem is one of the legs of a right triangle (see figure 3), the other leg being the value of ℓ / 2 (half of a side) and the hypotenuse the radius r of the polygon.
When the Pythagorean theorem is applied to said triangle, this equation is obtained, which is valid not only for the hexagon, but for any regular polygon.
It is the angle whose vertex coincides with the center O and whose sides are the segments that join the center with two consecutive vertices. Its measure in sexagesimal degrees is 360º / n, where n is the number of sides of the polygon.
It is the difference between the radius of the polygon and the apothem (see figure 3). Denoting the sagitta as S:
S = r - a
It is easily calculated by adding the lengths of the sides. Since any side has equal length L and there are n sides, the perimeter P is expressed as:
P = n.L
In a regular polygon the area A is given by the product between the semi-perimeter (half of the perimeter) and the length of the apothem to.
A = P.a / 2
Since the perimeter depends on the number of sides n, it turns out that:
A = (nL) .a / 2
Two regular polygons can have the same perimeter even if they do not have the same number of sides, since it would then depend on the length of the sides.
In book V of his Collection, the mathematician Pappus of Alexandria (290-350), the last of the great ancient Greek mathematicians, showed that among all the regular polygons with the same perimeter, the one with the greatest area is the one with the greatest number of sides.
Figure 4 shows the relevant angles in a regular polygon, denoted by the Greek letters α, β and γ.
Previously we mentioned the central angle, between the elements of the regular polygon, it is the angle whose vertex is in the center of the polygon and the sides are the segments that join the center with two consecutive vertices.
To calculate the measure of the central angle α, divide 360º by n, the number of sides. Or 2π radians between n:
α = 360º / n
Equivalent in radians to:
α = 2π / n
In figure 4 the internal angle β is the one whose vertex coincides with one of the figure and its sides are sides of the figure as well. It is calculated in sexagesimal degrees by:
β = [180 (n-2)] / n
Or in radians using:
β = [π (n-2)] / n
They are denoted by the Greek letter γ. The figure shows that γ + β = 180º. Therefore:
γ = 180º - β
The sum of all the external angles to a regular polygon is 360º.
Next we have the first 8 regular polygons. We observe that as the number of sides increases, the polygon increasingly resembles the circumference in which they are inscribed.
We can imagine that by making the length of the sides smaller and smaller, and increasing the number of these, we get the circumference.
Regular polygons are found everywhere in everyday life and even in nature. Let's see some examples:
Regular polygons such as equilateral triangles, squares and rhombuses abound in the signage we see on highways and roads. In figure 6 we see a stop sign with an octagonal shape.
Countless pieces of furniture have the square, for example, as a characteristic geometric figure, just as many tables, chairs and benches are square. A parallelepiped is generally a box with sides in the shape of a rectangle (which is not a regular polygon), but they can also be made square..
The tiles on floors and walls, both in homes and on the streets, are often shaped like regular polygons..
Tessellations are surfaces covered entirely with tiles that have different geometric shapes. With the triangle, the square and the hexagon you can make regular tessellations, those that only use a single type of figure to cover perfectly, without leaving empty spaces (see figure 6).
Likewise, buildings make use of regular polygons in elements such as windows and decoration..
Surprisingly, the regular hexagon is a polygon that appears frequently in nature..
The honeycombs made by bees to store honey are shaped very close to a regular hexagon. As Pappus of Alexandria observed, in this way the bees optimize the space to store as much honey as possible..
And there are also regular hexagons in the shell of the turtles and the snowflakes, which also adopt various very beautiful geometric shapes..
A regular hexagon is inscribed in a semicircle of radius 6 cm, as shown in the figure. What is the value of the shaded area?
The shaded area is the difference between the area of the semicircle with radius R = 6 cm and the area of the entire hexagon, a regular 6-sided polygon. So we will need formulas for the area of each of these figures.
TO1 = π Rtwo / 2 = π (6 cm)two / 2 = 18π cmtwo
The formula to calculate the area of a regular polygon is:
A = P.a / 2
Where P is the perimeter and to is the apothem. Since the perimeter is the sum of the sides, we will need the value of these. For the regular hexagon:
P = 6ℓ
Therefore:
A = 6ℓa / 2
To find the value of the side ℓ, it is necessary to construct auxiliary figures, which we will explain below:
Let's start with the small right triangle on the left, whose hypotenuse is ℓ. An internal angle of the hexagon is equal to:
α = [180 (n-2)] / n = α = [180 (6-2)] / 6 = 120º
The radius that we have drawn in green bisects this angle, therefore the acute angle of the small triangle is 60º. With the information provided, this triangle is solved, finding the light blue side, which measures the same as the apothem:
Opposite leg = a = ℓ x sin 60º = ℓ√3 / 2 cm
This value it's the double of the dark blue leg of the large triangle to the right, but from that triangle we know that the hypotenuse measures 6 cm because it is the radius of the semicircle. The remaining leg (bottom) is equal to ℓ / 2 since point O is in the middle of the side.
Since internal angles of this triangle are not known, we can state the Pythagorean theorem for it:
36 = 3 ℓtwo + ℓtwo / 4
(13/4) ℓtwo = 36 → ℓ = √ (4 x36) / 13 cm = 12 / √13 cm
With this value the apothem is calculated:
a = ℓ√3 / 2 cm = (12 / √13) x (√3 / 2) cm = 6√3 / √13 cm
Let's calltwo to the area of the regular hexagon:
= 28.8 cmtwo
TO1 - TOtwo = 18π cmtwo - 28.8 cmtwo = 27.7 cmtwo
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