In article <firstname.lastname@example.org>, 130bcd <email@example.com> wrote:
> .... I have unsuccessfully tried to find the perimeter of a pentagon. The "radius" > to the vertices is 130, which I tried to convert into 5 triangles but I can't > find a formula for determining the length of the 3rd side of an isosceles > triangle. > > The two equal length sides would, of course be 130 and the angle between them > would be 72....
If a regular n-gon is inscribed in a circle of radiu r, then each side of the n-gon has length 2r.sin(pi/n). In your case that means 260.sin(pi/5) or 260.sin(36 degrees).
That can actually be expressed in terms of square roots if you like, since sin(36 degrees) = (1/4)sqrt(10 - 2.sqrt(5)). Such expressions involving sqrt(5) underlie the construction of the regular pentagon by straightedge and compasses (Euclid IV.11, or a neater construction by Ptolemy).