Degree of curvature and radius relation

Degree of curvature $D$ is the central angle subtended by an arc (or chord) of standard length $C$ along the curve; for metric work $C = 30$ m. It describes the sharpness of the curve: a bigger $D$ means a sharper curve and smaller radius.

Derivation

By definition, an arc of length $C$ subtends the angle $D$ at the centre. In a circle, arc = radius × angle (radians):

$$C = R\times D\times\frac{\pi}{180}$$

Therefore

$$R = \frac{180\,C}{\pi D}$$

For $C = 30$ m:

$$R = \frac{180\times 30}{\pi D} = \frac{1718.87}{D}\ \text{m}\qquad\text{or}\qquad D = \frac{1718.87}{R}$$

For $C = 100$ ft, $R = 5729.58/D$ feet.

If the degree is defined by a chord, $R = \dfrac{C}{2\sin(D/2)}$; for flat curves this is almost identical to the arc definition.

Example: $D = 4^\circ$ gives $R = 1718.87/4 = 429.72$ m; a 6° curve gives $R = 286.48$ m.