The value 4.47204 from Geometers' Sketchpad could be R*2*sqrt(5). (That identification of the number 4.47204 comes from A Dictionary of Real Numbers, by Jonathan Borwein and Peter Borwein, Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, California, 1990.)

Via calculus we want to maximize 2*sin(x) + 4*cos(x), leaving R constant of course:

Its derivative is 2*cos(x) - 4*sin(x)

Equate this to zero so that

cos(x)=2*sin(x)

tan(x)=1/2

x=26.5650511771 degrees

2*sin(x)+4*cos(x)=4.472135955

That value squared = 20, whose square root is indeed 2*sqrt(5).

So the maximum perimeter is in fact R*2*sqrt(5).