# Degree of curvature

Degree of curve or degree of curvature is a measure of curvature of a circular arc used in civil engineering for its easy use in layout surveying.

## Definition

The degree of curvature is defined as the central angle to the ends of an arc or chord of agreed length.[1] Various lengths are commonly used in different areas of practice. This angle is also the change in forward direction as that portion of the curve is traveled.

## Usage

Curvature is usually measured in radius of curvature. A small circle can be easily laid out by just using radius of curvature, But if the radius is large as a km or a mile, degree of curvature is more convenient for calculating and laying out the curve of large scale works like roads and railroads. By this method curve setting can be easily done with the help of a transit or theodolite and a chain, tape or rope of a prescribed length.

A n-degree curve turns the forward direction by n degrees over Arc (or chord) length. The usual distance in US road work is 100 ft (30.48 m) of arc.[2] US railroad work traditionally used 100 ft of chord and this may be used in other places for road work. Other lengths may be used—such as 30 metres or 100 metres where SI is favoured, or a shorter length for sharper curves. Where degree is based on 100 units of arc length, the conversion between degree and radius is DR = 5729.57795, where D is degree and R is radius.

Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential and this made work easier before electronic calculators became available.

The 100 ft (30.48 m) is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00 etc. Metric work may use similar notation, such as kilometers plus meters 1+000.

## Formula

Degree of curvature can be converted to radius of curvature by the following formulae:

Diagram showing different parts of the curve used in the formula

### Formula for Arc length

$R = \frac{A*180}{D_A*\pi}$

here

$A$ is arc length
$R$ is radius of curvature
$D_A$ is degree of curvature, arc definition

### Formula for Chord length

$R = \frac{\frac{C}{2}}{\sin (\frac{D_C}{2})}$

here

$C$ is chord length
$R$ is radius of curvature
$D_C$ is degree of curvature, chord definition