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
Angle of thumb approximations only. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—"amphicyrtic" (Gr. , on both sides, κυρτός, convex) or "cissoidal" (Gr. κισσός, ivy), biconvex; "xystroidal" or "sistroidal" (Gr. ξυστρίς, a tool for scraping), concavo-convex; "amphicoelic" (Gr. κοίλη, a hollow) or "angulus lunularis", biconcave. The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only