---
title: "Degree of curvature and radius relation"
subject: "Surveying II"
chapter: 6
key: "semester_4/surveying_ii:degree-of-curvature-radius-relation"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/degree-of-curvature-radius-relation
site: BCT Engineering
---

# Degree of curvature and radius relation

Surveying II · Chapter 6: Curves

- Asked in: 2081 Chaitra, 4 marks
- Video: [Part 7 ! Curve ! Curve Lecture ! Surveying Lecture ! Relation Between Degree and Radius of a Curve !](https://www.youtube.com/watch?v=Zdsa7AByhu8) — NARAYANA COACHING CLASSES
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/degree-of-curvature-radius-relation
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-6

## Question

What is degree of curvature? Derive the relation between the radius and degree of curvature with its finding.

## Short answer

Degree of curvature D is the central angle subtended by a standard arc (usually 30 m). The radius is R = 180C/(πD); for C = 30 m, R = 1718.87/D m. A 4° curve gives R ≈ 429.72 m.

## Answer

**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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Degree of curvature is the central angle subtended by
   A. the whole curve
   B. the tangent length
   C. a standard arc length C (30 m metric)
   D. the long chord
   Answer: C. D is defined for a standard arc (or chord) of length C along the curve. (Key point: Definition)

2. Which relation is the basis of the derivation?
   A. C = R tan D
   B. C = R/D
   C. C = 2R sin D
   D. C = R D (π/180), giving R = 180C/(πD)
   Answer: D. Arc length = radius x angle in radians, so R = 180C/(πD). (Key point: Derivation)

3. For a 30 m arc, a 4° curve has radius
   A. 429.72 m
   B. 7.5 m
   C. 1718.87 m
   D. 286.48 m
   Answer: A. R = 1718.87/4 = 429.72 m. (Key point: Numerical)

4. If D doubles, the radius
   A. quadruples
   B. halves
   C. doubles
   D. stays the same
   Answer: B. R is inversely proportional to D. (Key point: Effect on radius)

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