Octagon eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° (formula_1 radians). The central angle is 45° (formula_2 radians). The area of a regular octagon of side length "a" is given by In terms of the circumradius "R", the area is In terms of the apothem "r" (see also inscribed figure), the area is These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as where
Octagon be seen in the vertices and side edges of a square antiprism with the same D, [2,8] symmetry, order 16. The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A, B, and D Coxeter planes. The "regular octagon" has Dih symmetry, order 16. There are 3 dihedral subgroups: Dih, Dih, and Dih, and 4 cyclic subgroups: Z, Z, Z, and Z, the last implying no symmetry. On the regular octagon, there are 11 distinct symmetries. John Conway labels full symmetry as r16. The dihedral symmetries are