Sorry if this is too elementary, if $R$ is the radius how do I visualize $1/R$? Thanks.
$1/R$ can be seen in many different ways.
One way to think of $1/R$ is that it is the curvature of the circle (seen as a curve in the plane).
If you wish to "draw" the length $1/R$ by using straightedge-compass constructions, it is possible to "look" at $1/R$, too. I'll shoot a picture of this here :
You start with a circle of radius $R$, and then draw a tangent line segment to the circle that has length 1 in both directions from the tangent point. This gives you the triangle formed next, and you can use compass & straightedge construction to draw the lines perpendicular to the triangle sides that goes through the point lying on the middle of the sides. These three lines intersect at the center of a circle that goes through all three vertices of this triangle, and there is a theorem in Euclidean geometry that says that if two straight lines go through a circle, we have $$ ac = bd $$ (in my drawing, we could replace the straightlines by any straightlines, and the roles of $R,1/R,1,1$ could be replaced by $a,c,b,d$, respectively, and the intersection needs not to be orthogonal). Therefore, the length that I pointed in the drawing to be $1/R$, call it $x$, satisfies $$ Rx = 1 \cdot 1 = 1 $$ thus $x = 1/R$.
Hope that helps!