The Perimeter of a Polygon (Outer Radius) calculator computes the length of the perimeter of a regular polygon of (n) sides that is circumscribed by an outer circle of radius (r).
INSTRUCTIONS: Choose units and enter the following:
Polygon Perimeter (P): The calculator return to total perimeter of the regular polygon in meters. However this can be automatically converted to compatible units via the pull-down menu.
A regular n-sided polygon is a polygon with n equal length sides and is a polygon which has n equal angles at the n vertices of the polygon. Because of the symmetry of this construction, all the vertices of the regular polygon lie on the circle and the sides of the regular polygons form n chords of the circle. The formula for the perimeter of a polygon based on the number of sides and the outer radius is:
P = 2⋅n⋅r⋅sin(π/n)
The n-sided area of a regular polygon, as can be seen in Figure 1, is comprised of n isosceles triangles. The regular polygon is constructed to have all its vertices on the circle, and thus a radii of the circle intersects a vertex of the polygon and bisects the angle of the polygon. It also can be shown that the radius to the polygon's vertex forms the side of n triangles whose third side is the polygon's side. Since these triangles with base s are isosceles triangles, the radii intersecting the polygon's side at s/2 splits these triangles into two right triangles .
If we imagine the sides of the polygon as the bases of these isosceles triangles, then we can see the area of each of the n triangles is given by the simple formula:
 `"Perimeter"_"(Polygon)" = n * s`
We can find the length of the polygons side s by noting first that the triangle with base s/2 and height L is a right triangle. We also note that the angle, `alpha`, is given by:
 `alpha = (2 * pi) /n`, because all the n equal angles `alpha` must sum to `2pi` radians
We also see that the circle's radius, r, is the hypotenuse of a right triangle and thus relates r and s/2 as:
 `s/2 = r * sin (alpha/2)`
Substituting equation  into equation  we get:
 `s/2 = r *sin( ((2*pi) / n)/2 )`
And rearranging we get the polygon side, s, in terms of the circle's radius, r, and the number of sides of the regular polygon, n:
 `s = 2 * r * sin( pi / n)`
We finally compute the total length of the polygon's perimeter by substituting equation  into equation :
 `"Perimeter"_"(Polygon)" = n * 2 * r * sin( pi / n) `