Abstract
A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon PN. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.
Highlights
A simple polygon. . . that has equal all sides or equal all interior angles is called a semi-regular polygon [8, 9, 10, 11]
We distinguish between two types of semi-regular polygons: equiangular and equilateral
Interior angles of a semi-regular polygon at odd vertices are marked with, and those at even vertices are marked with
Summary
That has equal all sides or equal all interior angles is called a semi-regular polygon [8, 9, 10, 11]. We distinguish between two types of semi-regular polygons: equiangular (having equal interior angles and different sides) and equilateral (having equal sides, and different interior angles). Of a polygon is in an even position, or odd position, if index is even, or odd number, respectively. In this paper we consider convex equilateral semi-regular polygons. Is a number of sides in a semi-regular equilateral polygon, 2. Is a number of sides in a regular polygon, 3. Is a side in a semi-regular polygon , 4. Interior angles of a semi-regular polygon at odd vertices are marked with , and those at even vertices are marked with
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