Editing Octagon (section)

=== Area ===
The area of a regular octagon of side length ''a'' is given by
:<math>A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \approx 4.828\,a^2.</math>

In terms of the [[Circumscribed circle|circumradius]] ''R'', the area is
:<math>A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \approx 2.828\,R^2.</math>

In terms of the [[apothem]] ''r'' (see also [[inscribed figure]]), the area is
:<math>A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \approx 3.314\,r^2.</math>

These last two [[coefficients]] bracket the value of [[pi]], the area of the [[unit circle]].
[[File:Octagon in square.svg|frame|The [[area]] of a [[Regular polygon|regular]] octagon can be computed as a [[Truncation (geometry)|truncated]] [[Square (geometry)|square]].]]

The area can also be expressed as
:<math>\,\!A=S^2-a^2,</math>

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 overlap with the four sides of the square) and then takes the corner triangles (these are [[Special right triangles#45–45–90 triangle|45–45–90 triangles]]) and places 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
:<math>S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a \approx 2.414a.</math>
The span, then, is equal to the ''[[silver ratio]]'' times the side, a.

The area is then as above:
:<math>A=((1+\sqrt{2})a)^2-a^2=2(1+\sqrt{2})a^2 \approx 4.828a^2.</math>

Expressed in terms of the span, the area is
:<math>A=2(\sqrt{2}-1)S^2 \approx 0.828S^2.</math>

Another simple formula for the area is 
:<math>\ A=2aS.</math>

More often the span ''S'' is known, and the length of the sides, ''a'', is to be determined, as when cutting a square piece of material into a regular octagon. From the above,
:<math>a \approx S/2.414.</math>

The two end lengths ''e'' on each side (the leg lengths of the triangles (green in the image) truncated from the square), as well as being <math>e=a/\sqrt{2},</math> may be calculated as
:<math>\,\!e=(S-a)/2.</math>