We will verify that the circular functions of the reals $x = \dfrac{\pi}{n}$, with $n \in \mathbb{N}$ and $n \ge 3$, can be calculated from $\ell_n$, the side of the regular polygon with $n$ sides inscribed in the cycle. The story In book I of Almagest, Ptolemy constructed his table of chords exactly like this: he started from the sides of inscribed regular polygons — triangle, square, pentagon, hexagon and decagon —, which Euclid’s geometry knew how to calculate, and obtained the other arcs through formulas for sum, difference and half of arcs, which we will see in chapter VI. The chord of an arc $2\alpha$ in a circle of radius $R$ is $2R\operatorname{sen}\alpha$: the theorem in this chapter is the modern translation of this idea.
16 Capítulo 16 · Neste capítulo você vai Regular polygons recognize regular polygons and prove that they are inscriptible and circumscriptible; identify center, apothem and centric angle; calculate side and apothem of the square, hexagon and equilateral triangle as a function of the radius; obtain the side of the regular decagon and pentagon and recognize the golden segment. Hexagonal nuts, honeycombs, the STOP sign, the gears of a clock: regular polygons are where you need to repeat the same piece around a center. In this chapter, we prove that every regular polygon fits exactly between two circles with the same center and we calculate its sides and apothem from the radius — calculations that, almost two thousand years ago, were used to create the first trigonometric tables.
In previous posts we talked about Stewart’s theorem, here we will talk about the circumference relationship in regular polygons. The relationship between circles and regular polygons is a fascinating and complex topic in geometry. Regular polygons are geometric figures with congruent sides and angles, and the circle can be seen as a limiting case of a regular polygon with an infinite number of sides. When a regular polygon is inscribed in a circle, each of its vertices touches the circle, and the center of the polygon coincides with the center of the circle. This allows precise relationships to be established between the radius of the circle and the dimensions of the polygon, such as the side and apothem (the distance from the center to a side).
In previous posts we talked about some concepts about polygons in general. Regular polygons are closed, flat and fascinating geometric figures that have aroused the interest of mathematicians and artists throughout history. Characterized by having all sides and internal angles congruent, these figures are the embodiment of symmetry and harmony. A flat geometric figure with sides of equal length and internal angles of equal measure. This means that a regular polygon not only looks balanced, but also follows precise mathematical rules that can be expressed through elegant and useful formulas.