Prototype railroads in North America measure curves by degree of curvature, not radius. The degree of curvature is measured by in the field by the offset of a 100 foot chord.
For a prototype curve:
R=Radius of the curve
D=Degree of curvature
So for a 4 degree 30 minute curve:
R= 50/sin(4 deg 30 min/2)
R= 50/sin(2 deg 15 min)
R= 1275 feet
To convert to HO radius in inches:
Rho= 175 inches
Typical mainline curves are in the 2-6 degree range. Sharp curves are those above 10 degrees.
In the United Kingdom and most countries that belonged to the former British Empire, prototype curves are commonly measured by radius. In pre-metric times, the radius was often measured in chains.
1 chain = 22 yards = 66 feet.