Regular Polygons: Properties, Elements, Angles, Examples
The regular polygons are those with all sides and 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.
Figure 1. Regular polygons are those whose sides and internal angles are equal, such as the equilateral triangle and the square. Source: Wikimedia Commons.
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 interior 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.
Article index

one
Properties of regular polygons

two
Elements of a regular polygon

2.1
Vertex

2.2
Side

23
Diagonal

2.4
Center

2.5
Radio

2.6
Apothem

2.7
Central angle

2.8
Sagita


3
Perimeter and area

3.1
Perimeter

3.2
Area


4
Angles

4.1
Central angle

4.2
Internal angle or internal angle

4.3
External angles


5
Examples of regular polygons

5.1
– Regular polygons in daily life and nature

5.2
– Regular hexagons in nature


6
Exercise resolved

6.1
Solution


7
References
Properties of regular polygons
Properties of regular polygons
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 their 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 a circumscribed circumference .
For a regular polygon with n sides, the measure of an interior angle α is:
α = [180 (n2)] / n
You can draw n (n3) / 2 diagonals from the vertices of a polygon, whether regular or not.
The sum of the exterior angles is equal to 360º.
Figure 2. Inscribed circumference and circumference circumscribed to a regular polygon. Source: F. Zapata.
Elements of a regular polygon
Elements of a regular polygon
Next we present the main elements of a regular polygon, visualized in the figure below.
Figure 3. Elements of the regular polygon. Source: F. Zapata.
Vertex
Vertex
Common point that two consecutive sides have, denoted as V in the figure.
Side
Side
It is the segment that joins two consecutive vertices of the polygon and is denoted as ℓ or L.
Diagonal
Diagonal
Segment that joins two nonconsecutive vertices of the polygon, in the figure it is denoted as d .
Center
Center
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.
Radio
Radio
It is the radius r of the circumscribed circle and coincides with the distance between O and a vertex.
Apothem
Apothem
The apothem is called the radius of the circumference inscribed in the polygon, represented in the figure with a letter a . 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.
Central angle
Central angle
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.
Sagita
Sagita
It is the difference between the radius of the polygon and the apothem (see figure 3). Denoting the sagitta as S:
S = r – a
Perimeter and area
Perimeter and area
Perimeter
Perimeter
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 = nL
Area
Area
In a regular polygon the area A is given by the product between the semiperimeter (half of the perimeter) and the length of the apothem a .
A = Pa / 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 (290350), the last of the great ancient Greek mathematicians, showed that among all the regular polygons with the same perimeter, the one with the largest area is the one with a greater number of sides.
Angles
Angles
Figure 4 shows the relevant angles in a regular polygon, denoted by the Greek letters α, β and γ.
Central angle
Central angle
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
Internal angle or internal angle
Internal angle or internal angle
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 (n2)] / n
Or in radians using:
β = [π (n2)] / n
External angles
External angles
They are denoted by the Greek letter γ. The figure shows that γ + β = 180º. Therefore:
γ = 180º – β
The sum of all the exterior angles to a regular polygon is 360º.
Figure 4. The angles in a regular polygon, in this example a regular pentagon. Source: Wikimedia Commons.
Examples of regular polygons
Examples of regular polygons
Next we have the first 8 regular polygons. We observe that as the number of sides increases, the polygon becomes more and more like 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 in daily life and nature
– Regular polygons in daily life and nature
Regular polygons are found everywhere in everyday life and even in nature. Let’s see some examples:
Traffic signals
Regular polygons such as equilateral triangles, squares and rhombuses abound in the signage we see on highways and roads. In figure 6 we see an octagonal stop sign.
Figure 5. Octagonal shaped traffic sign. Source: Pixabay.
Furniture
Countless pieces of furniture have the square, for example, as their 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.
Architecture and construction
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 various geometric shapes. With the triangle, the square and the hexagon, regular tessellations can be made, those that only use a single type of figure to cover perfectly, without leaving empty spaces (see figure 6).
Likewise, the buildings make use of the regular polygons in elements such as windows and decoration.
Figure 6. Square tile floor. Source: Pixabay.
– Regular hexagons in nature
– Regular hexagons in nature
Surprisingly, the regular hexagon is a polygon that appears frequently in nature.
The combs made by bees to store honey are shaped very roughly to a regular hexagon. As Pappus of Alexandria observed, in this way bees optimize space to store as much honey as possible.
And there are also regular hexagons in the shells of turtles and snowflakes, which also take on various very beautiful geometric shapes.
Exercise resolved
Exercise resolved
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?
Figure 7. A regular hexagon inscribed in a semicircle. Source: F. Zapata.
Solution
Solution
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 6sided polygon. So we will need formulas for the area of each of these figures.
Semicircle area
A _{1} = π R ^{2} /2 = π (6 cm) ^{2} /2 = 18π cm ^{2}
Regular hexagon area
The formula to calculate the area of a regular polygon is:
A = Pa / 2
Where P is the perimeter and a 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 (n2)] / n = α = [180 (62)] / 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 is twice 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 ℓ ^{2} + ℓ ^{2} /4
(13/4) ℓ ^{2} = 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 call A _{2} the area of the regular hexagon:
= 28.8 cm ^{2}
Shaded figure area
A _{1} – A _{2} = 18π cm ^{2} – 28.8 cm ^{2} = 27.7 cm ^{2}
References
References

Baldor, A. 1973. Geometry and trigonometry. Central American Cultural Publishing House.

Enjoy math. Tessellations. Recovered from: enjoymatematicas.com.

EA 2003. Elements of geometry: with exercises and geometry of the compass. University of Medellin.

Hexagons in nature. Recovered from: malvargamath.wordpress.com.

Jiménez, R. 2010. Mathematics II. Geometry and trigonometry. Second edition. Prentice Hall.

Regular polygons. Recovered from: mate.ingenieria.usac.edu.gt.

Wikipedia. Apothem. Recovered from: es.wikipedia.org.