Arc length

Definition

The arc length on the part of a circle.

Arc length is the distance along the arc/curve,^[1] denoted by "S", the target arc can be denoted by"L" or "[math]\displaystyle{ \overset{\frown}{MN} }[/math]", not the distance between its endpoints, which is known as the chord length and is denoted by "C".

Not all curves can be defining length, curves that can be defined by the length are called the rectifiable curves.^[2]

Calculate the Arc length on a circle

Depending on the given, there are two different ways to find the Arc Length.^[3]

With the given of Center Angle & Radius

Arc length on a circle can be measured as Arc length = s = θ/360 of 2πr = θ/360 × 2πr = rθ × π/180. (θ represents the degree of the Center angle,"r" represents the Radius of the circle).

With the given of Curvature & Radius

Arc on circle

Arc length on a circle can be measured as Arc length= s = r × θ

calculate the arc length

Calculate the Arc length is not on a circle

The original method for calculating the arc length involves dividing the arc into many parts to obtain an estimated answer:

The curve is divided into smaller segments, and the length of each segment can be calculated by the Pythagorean theorem or other methods.
The total length of the curve can be obtained by summing each part.

Arc Length Media

When rectified, the curve gives a straight line segment with the same length as the curve's arc length.
Arc length s of a logarithmic spiral as a function of its parameter θ.
Approximation to a curve by multiple linear segments, called rectification of a curve.
quarter circle
Fermat's method of determining arc length
Koch curve in seven iterations.
The graph of x \cdot \sin(1/x)