Octagon - Regular Octagon

Regular Octagon

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon). The area of a regular octagon of side length a is given by

In terms of R (circumradius), the area is

In terms of r (inradius), the area is

These last two coefficients bracket the value of pi, the area of the unit circle.

The area can also be derived as follows:

where S is the span of the octagon, or the second shortest diagonal; and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45–45–90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the length of a side a, the span S is:

The area is then as above:

Expressed in terms of the span, area is:

Another simple formula for the area is

where d is the distance between parallel sides (the same as span S in the diagram).