Arc length is a term used in mathematics to refer to the distance along a curve or arc. It is a fundamental concept in geometry and calculus, and is used to measure the length of curves in two-dimensional and three-dimensional space. The concept of arclength is used in a wide range of fields, including engineering, physics, and computer graphics.

## Definitions

Arc length is defined as the length of a curve or arc, measured along the curve. It is the distance between two points on the curve, and is calculated by integrating the distance formula along the curve. The formula for arclength is given by:

L = ∫√(1+(dy/dx)²) dx.

Where L is the length of the curve, and dy/dx is the derivative of the curve.

## Origin

The concept of arclength has been used in mathematics for thousands of years. The ancient Greeks were the first to study the properties of curves, and developed the concept of arclength to measure the length of curves. The term “arclength” itself comes from the Latin word “arcus”, meaning “bow” or “curve”.

## Meaning in different dictionaries

The meaning of arclength is consistent across different dictionaries, with all defining it as the length of a curve or arc. The Oxford English Dictionary defines arclength as “the length of an arc of a curve”. The Merriam-Webster Dictionary defines it as “the length of a curve or arc”.

## Associations

The concept of arclength is closely associated with the study of curves and geometry. It is used to measure the length of curves in two-dimensional and three-dimensional space, and is a fundamental concept in calculus. Arclength is also used in engineering and physics, where it is used to calculate the distance traveled by a moving object along a curved path.

## Synonyms

Some synonyms of arclength include curve length, arc distance, and curve distance.

## Antonyms

There are no antonyms for arclength, as it is a specific term used to describe the length of a curve or arc.

## The same root words

The root word of arclength is “arcus”, which means “bow” or “curve”. Other words derived from this root include arc, arch, and archery.

## Example Sentences

- To find the arclength of a curve, you need to integrate the distance formula along the curve.
- The engineer used the arclength formula to calculate the distance traveled by the train along the curved track.
- The arclength of a circle is equal to the circumference of the circle.