I don't know anything about SketchUp but here's a construction of the Golden Rectangle that may help you scale the radius of the circle.
Construct a square and bisect the bottom edge.
Produce the bottom edge.
Draw an arc as shown to determine the length of the bottom edge of the Golden Rectangle.
No engineering or math wizardry ... just good old Euclidean geometry.
EDIT: The radius is the diagonal of a rectangle with the ratio of the sides being 2:1.
The angle measured between the radius and the produced edge = arctan 2 = 63.43495°.
