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A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. [1] Its Schläfli symbol is {10} [ 2 ] and can also be constructed as a truncated pentagon , t{5}, a quasiregular decagon alternating two types of edges.
A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side , then also the presentation is succeed by extension one side and it generates a segment, here F E 2 ¯ , {\displaystyle {\overline {FE_{2}}}{\text{,}}} which is divided according to the golden ratio:
In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of an octagonal antiprism with the same D 8d, [2 +,16] symmetry, order 32. The octagrammic antiprism , s{2,16/3} and octagrammic crossed-antiprism , s{2,16/5} also have regular skew octagons.
Three squares of sides R can be cut and rearranged into a dodecagon of circumradius R, yielding a proof without words that its area is 3R 2. A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12.
The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR 2 − 1 / 4 ns 2, where s is the side length and R is the circumradius. [4]: p. 73
The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a 2 + b 2 = c 2, so line segments with these lengths form a right triangle (by the converse of the Pythagorean theorem). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle. [8]
The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime , being of the form 2 2 n + 1 (in this case n = 4).
Close approximations to the regular hendecagon can be constructed. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long. [7] The hendecagon can be constructed exactly via neusis construction [8] and also via two-fold origami. [9]