Convex and regular polygons they are classifications of these geometric figures in relation to their shape. For a better understanding of these classificatory concepts, it is necessary to know some other basic concepts about polygons.
One polygon it is a region of the plane formed by the union of a closed line – which, in turn, is formed by straight segments called sides – and all the points interior to that line.
Examples of polygons are triangles, squares, rectangles and parallelograms. In addition to them, all geometric figures that follow the construction pattern of these examples are also polygons, such as pentagons, hexagons, heptagons, etc.
examples of polygons
They are not polygons, therefore, figures that present on one of their sides, instead of a line segment, any curve or that two of their sides intersect.
Examples of non-polygons
One polygon is convex when, given any two points A and B within it, it is impossible to find a segment of line AB with at least one point outside the polygon, the
that is, taking two points A and B within a polygon, if segment AB is always entirely inside the polygon, regardless of the location of points A and B, this polygon will be convex.
Examples of convex and non-convex polygons
In the image above, notice that polygon S has a kind of “mouth” between points C and E. Also note that point D advances towards the interior of the polygon. This polygon is not convex, a fact that can be noticed by the highlighted part of the AB segment. This part is outside the polygon, while points A and B are inside it. As defined above, polygon S is not a convex polygon.
With respect to polygon T, any location observed for points A' and B' generates a straight line segment A'B' totally interior to the polygon. Therefore, the T polygon is convex.
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Regular polygons are convex polygons that have all sides congruent and all interior angles congruent. Importantly, the angles and sides do not have to be the same measurement – claiming they have the same measurement doesn't even make sense. So the definition usually says "congruent sides and congruent internal angles” to avoid this kind of confusion.
Thus, any polygon where all sides and angles have the same measurement is called a regular polygon.
Examples of regular and non-regular polygons
In the image above, the polygon S is regular because it conforms to the definition. On the other hand, the T polygon is not regular. Although the figure looks like a regular polygon, one side of this polygon has a different measure than the others.
Any polygon has the following elements:
1 – sides: line segments constituting the contour of a polygon;
2 – vertices: meeting points between the sides.
A convex polygon, in addition to the elements mentioned above, has the following elements:
3 – Internal angles:angles formed by two consecutive sides in the interior region of the polygon.
4 – Outside angles: are formed by one side and the extension of the side following it. In this way, the sum of an inner angle and an outer angle belonging to the same vertex is always equal to 180°.
5 – diagonals: line segments that connect two non-consecutive vertices of a polygon.
Examples of the elements of a convex polygon
In the image above, the vertices are points A, B, C, D and E. The sides are AB, BC, CD, DE and EA. Diagonals are dotted lines. At vertex A, α is the inner angle and β is the outer angle.
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What are convex and regular polygons?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-sao-poligonos-convexos-regulares.htm. Accessed on June 27, 2021.
