# What is an Arc of a Circle?(Definition, Formulas & Examples)

A circle is a figure formed by the locus of a point moving in a plane, in such a way that its distance from a fixed point is always constant. The fixed point is called the centre of a circle and the constant distance between any point on the circle and its centre is called the radius. There are many parts in a circle such as radius, diameter, circumference, tangent, secant, chord, arc, segment, and sector.

Let’s understand what is an arc of a circle and what are the different formulas related to it.

## What is an Arc of a Circle?

In Mathematics, an arc means, a part of a curve or the portion of a circle. An arc of a circle is defined as the part or segment of the circumference of a circle. The straight line joining the ends of the arc is called the chord of the circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.

The arc is represented by the symbol ‘⌢’. The arc in the above figure is called arc $\text{AB}$ or $\text{BA}$. It can also be expressed as $\overparen{\text{AB}} \quad$ or $\overparen{\text{BA}} \quad$.

The angle subtended by an arc at the centre is the angle formed between the two line segments joining the centre to the endpoints of the arc.

In the above figure, $\overparen{\text{AB}} \quad$ is the arc of the circle with centre O and $\angle \text{AOB}$ is an angle subtended by $\overparen{\text{AB}} \quad$ at the centre $\text{O}$. $\angle \text{AOB}$ is called central angle of an arc $\text{AB}$.

## What is Arc Length?

The arc length is defined as the interspace between the two points along a section of a curve. It is the length of an arc of a circle. The arc length of this arc $\overparen{\text{AB}} \quad$ is given as $\text{L}$.

## How to Find Arc Length of a Circle?

The arc length of a circle depends on the central angle of an arc. The central angle $\theta$ can be measured either in degrees or in radians.

When $\theta$ is measured in degrees, then arc length = $\frac{\pi r \theta}{180}$.

When $\theta$ is measured in radians, then arc length = $r \theta$.

## Derivation of Arc Length Formula

Circumference of a circle of radius $r$ is given by $2 \pi r$.

The angle subtended by a complete circle = $360^{\circ}$.

Therefore length of arc that subtends an angle of  $360^{\circ} = 2 \pi r$

$=>$ Length of arc that subtends an angle of  $1^{\circ} = \frac{2 \pi r}{360} = \frac{\pi r}{180}$

Thus the length of the arc that subtends an angle of  $\theta = \frac{\pi r \theta}{180}$.

As $1^{\circ} = \frac{180}{\pi}$

Therefore the length of the arc that subtends an angle $\theta$ measured in radians = $\frac{\pi r \theta}{180} \times \frac{180}{\pi} = r \theta$.

## Examples on Arc Length

Example 1: Find the length of an arc of a circle cut off by a central angle of 3 radians in a circle with a radius of 4 inches.

Central angle $\theta = 3$ radians

Radius of a circle $r = 4$ inches

Arc length = $r \theta = 4 \times 3 = 12$ inches

Example 2: The radius of the circle is $3.5$ cm and the arc subtends $75^{\circ}$ at the centre. What is the length of the arc?

Radius of circle $r = 3.5$ cm

Central angle $\theta = 75^{\circ}$

Arc length = $\frac{\pi r \theta}{180} = \frac{\frac{22}{7} \times 3.5\times 75}{180}$

$= \frac{\frac{22}{2} \times 75}{180}$

$= \frac{11 \times 75}{180} = 4.58$ cm

Example 3: Find the arc length of a circle with a radius measuring $5$ cm and a central angle of $120^{\circ}$. Express the answer in terms of $\pi$.

Radius of circle $r = 5$ cm

Central angle $\theta = 120^{\circ}$

Arc length = $\frac{\pi r \theta}{180} = \frac{\pi \times 5 \times 120}{180}$

$= \frac{10 \pi}{3}$

## Practice Problems

1. Find the arc length formed by the angle $\frac{\pi}{3}$ of a circle whose radius is a length of $5$ cm
2. Find the arc length formed by an angle measuring $30^{\circ}$ of a circle whose radius is a length of $3$ cm.
3. What is the measure of the angle that forms an arc with a length of $2.33$ cm of a circle that has a radius of $4$ cm?
4. An arc has a measure of $\frac{\pi}{4}$ and a diameter of $7$ cm, what is the measure of the central angle?

## FAQs

### What is arc of a circle?

The arc of a circle is defined as the length of a part of its circumference that lies between any two points on it. i.e., An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the endpoints of the arc.

### What is the arc length of a circle formula?

The arc length of a circle formula involves its radius ($r$) and the central angle ($\theta$). It is denoted by $\text{L}$ and is calculated by

a) the formula $\text{L} = \frac{\pi r \theta}{180}$ if $\theta$ is in degrees
b) the formula using $\text{L} = r \theta$ if $\theta$ is in radians

### How to calculate arc length using radians?

The arc length can be calculated when the central angle is given in radians using the arc of a circle formula: Arc Length = $r \theta$, when $\theta$ is in radians.

### What is the length of the major arc using the arc length formula?

A major arc in a circle is larger than a semicircle. Its central angle is larger than $180^{\circ}$. Using the formula $l = r \theta$ we can find the length of an arc of a circle, where $\theta$ is in radians.

## Conclusion

An arc of a circle is a part or segment of the circumference of a circle. The angle subtended by the end points of an arc at the centre of the circle is called the central angle. The arc length is calculated using the formula $\frac{\pi r \theta}{180}$ when the central angle $\theta$ is in degrees or  $r \theta$ when the central angle $\theta$ is in radians.