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
Content (measure theory) by This is an outer measure, in other words it has the following properties: The function μ above is an outer measure on the family of all subsets. Therefore it becomes a measure when restricted to the measurable subsets for the outer measure, which are the subsets "E" such that μ("X") = μ("X"∩"E") + μ("X"\"E") for all subsets "X". If the space is locally compact then every open set is measurable for this measure. The measure μ does not necessarily coincide with the content λ on compact sets, However it does if λ is regular in the sense that for