# Arc of a Circle

## What is an Arc of a Circle?

The length of a complete circle is the circumference of the circle. Now, if we take a section of the circumference, then the section may be called the Arc of the Circle.

Also, the section of a circle that is formed by cutting at any two points are called the Arc of the circle.

As shown in the figure, the circle is cut along the points $P$ and $Q$ and its circumference is divided into two sections represented by the colour Red and Blue. These two sections are called the Arcs of the circle.

## Symbol of Arc of a Circle

The symbol for representing an Arc is given by: $⌒$

## Angle Subtended by an Arc of a Circle

The angle made by the complete circumference of a circle with the centre is $360\degree$ or $2\pi$ radians. Since the Arc of a circle is a part of the circumference, therefore, the angle made by the Arc with the centre is less than $360\degree$ or $2\pi$ radians. The angle that the Arc makes with the Cente of the Circle as shown in the figure below is called the Angle Subtended by an Arc of a Circle.

As shown in the above figure, The Angle Subtended by the Arc represented by the colour Red is $\theta_2$ and the Angle Subtended by the Arc represented by the colour Blue is $\theta_1$.

## How to find the length of Arc of a Circle?

### Length of an Arc of a Circle in Degrees

The general formula for finding the Length of Arc of a Circle in Degrees is given by,

$L = 2\pi r\left( {\frac{\theta }{{360}}} \right)$

Where, $r$ is the radius of the circle, and $\theta$ is the angle in degrees subtended by the Arc with the centre of the circle.

### Length of an Arc of a Circle in Radians

The formula for finding out the length of an Arc in Radians is given by,

$L = \theta r$

Where, $r$ is the radius of the circle, and $\theta$ is the angle in radians subtended by the Arc with the centre of the circle.

## Major and Minor Arc

### Major Arc of a Circle

The Arc whose length is more than the half of the circumference of a circle is called Major Arc. The angle subtended by a Major Arc is always greater than $180\degree$ or $\pi$ radians.

In simple words when a circle is divided into two arcs the arc with longer length is called the Major Arc. The Major Arc is Generally Represented by using three Letters.

As shown in the above figure, the Major Arc is $POQ$.

### Length of a Major Arc of a Circle

From the above figure, the angle subtended by the Major Arc with the centre of the circle is $\theta_2$ and $r$ is the radius of the circle. Therefore, the length of the Major Arc $L_1$ in Degrees is given by,

${L_1} = 2\pi r\left( {\frac{{{\theta _2}}}{{360}}} \right)$

If $\theta_2$ is in radians, then the the formula for length of the Major Arc $L_1$ in Radians is given by,

${L_1} = {\theta _2}r$

### Minor Arc of a Circle

If the length of an Arc is less than the half of the circumference of a circle, then, the arc is called Minor Arc. The angle subtended by a Minor Arc with the centre of a circle is always less than $180\degree$ or $\pi$ radians.

In simple words, when a circle is divided into two arcs, the arc with shorter length is called the Minor Arc. The Minor Arc is generally Represented by using two Letters.

The above figure shows the Minor Arc is $PQ$.

### Length of a Minor Arc of a Circle

As shown in the figure, the angle subtended by the Minor Arc $PQ$ with the center of the circle is $\theta_1$. If $r$ is the radius of the circle, then the formula for finding the Minor Arc in Degrees is given by,

${L_2} = 2\pi r\left( {\frac{{{\theta _1}}}{{360}}} \right)$

If $\theta_1$ is in radians, then the formula for length of the Minor Arc $L_2$ in Radians is given by,

${L_2} = {\theta _1}r$

## Solved Examples on Arc of a Circle

1. The arc of a circle subtends an angle of $90\degree$ with the centre and the radius of the circle is $14$ cm. Find out whether the Arc is a Minor Arc or Major Arc and find the Arc Length in Degrees?

Solution: Since the Arc subtends an angle of $90\degree$ with the centre, therefore, the Arc is a Minor Arc.
Length of the Arc:
Given, $\theta=90\degree$ and $r=14$ cm.
$L = 2\pi r\left( {\frac{\theta }{{360}}} \right)$
$L = 2 \times \frac{{22}}{7} \times 14\left( {\frac{{90}}{{360}}} \right)$
$L = 2 \times 22 \times 2\left( {\frac{1}{4}} \right)$
$L = 22\,cm$
Therefore, the length of the Arc is $22 \,cm$.

2. The arc of a circle subtends an angle of $270\degree$ with the centre and the radius of the circle is $21$ cm. Find out whether the Arc is a Minor Arc or Major Arc and find the Arc Length in Degrees?

Solution: As the Arc subtends an angle of $270\degree$ with the centre of the circle which is greater than $180\degree$, therefore, the Arc is a Major Arc.
Length of the Arc:
Given, $\theta=270\degree$ and $r=21$ cm.
$L = 2\pi r\left( {\frac{\theta }{{360}}} \right)$
$L = 2 \times \frac{{22}}{7} \times 21\left( {\frac{{270}}{{360}}} \right)$
$L = 2 \times 22 \times 3 \times \frac{3}{4}$
$L = 11 \times 3 \times 3$
$L = 99\,cm$
Therefore, the length of the Arc is $99 \,cm$.

3. The arc of a circle subtends an angle of $\frac{\pi }{2}$ radians with the centre and the radius of the circle is $35$ cm. Then find out the Length of the Arc in Radians?

Solution: Given, $\theta=\frac{\pi }{2}$ radians and $r=35$ cm.
Length of the Arc in Radians:
$L = \theta r$
$L = \frac{\pi }{2} \times 35$
$L = 17.5\pi$
We know that, $\pi=3.1415$
$17.5\pi$ cm=$54.95$ cm
Therefore, the length of the Arc is $17.5\pi$ cm or $54.95$ cm.

4. Find the length of the Arc if the radius of the arc is $14$ cm and the angle subtended by the arc is $1.54$ radians?

Solution: Given, $\theta=1.54$ radians and $r=14$ cm
$L = \theta r$
$L=1.54\times 14$
$L=21.56\,cm$
Therefore, the length of the Arc is $21.56\,cm$.

5. If the length of an Arc is 24 cm and radius is 6 cm then, find the angle subtended the Arc in radians?

Solution: Given, $L=24\,cm$ and $r=6\,cm$
$L = \theta r$
$\theta = \frac{L}{r}$
$\theta = \frac{{24}}{6}$
$\theta = 4$ radians
Therefore, the angle subtended by the Arc is $4$ radians.

6. The angle subtended at the center of a circle of radius 18 cm by an arc 16.5cm long is?

Solution: Given, $L=16.5\,cm$ and $r=18\,cm$
$L = \theta r$
$\theta = \frac{L}{r}$
$\theta = \frac{{16.5}}{18}$
$\theta = 0.917$ radians
Therefore, the angle subtended by the Arc is $0.917$ radians.

7. If the measure of minor arc of circle is 125 degree then find the measure of major arc of the circle?

Solution: Given, Minor Arc say $L_1=125^0$ and let the Major Arc be $L_2$.
The total Arc of a Circle is $360^0$.
$\therefore\,\,\,\,\, L_1+L_2=360^0$
$\Rightarrow\,L_2=360^0-L_1$
$\Rightarrow\,L_2=360^0-125^0$
$\Rightarrow\,L_2=235^0$
Therefore, the Major Arc of the circle is $235^0$.

What is an Arc in a Circle?

The Arc in a Circle is a part of its circumference. If we cut a circle at any two points, the part of the circle obtained is called the Arc of the circle.

How do you find the Arc of a Circle?

The Arc can be found out by using the following formulas if we know the angle subtended by the arc and radius of the circle,
Arc $(L)$ if angle subtended is given in degrees: $L = 2\pi r\left( {\frac{\theta }{{360}}} \right)$
Arc $(L)$ if angle subtended is given in radians :$L = \theta r$

How do you define an Arc?

An Arc can be defined as a part of a circle obtained when it is cut at any two points.

What is the symbol for Arc?

The symbol for representing an Arc is given by: $⌒$

What is Major Arc?

The Arc whose length is more than the half of the circumference of a circle is called Major Arc. The angle subtended by a Major Arc is always greater than $180\degree$ or $\pi$ radians. In simple words when a circle is divided into two arcs then the arc with longer length is called the Major Arc.

What is Minor Arc?

If the length of an Arc is less than the half of the circumference of a circle, then, it is called Minor Arc. The angle subtended by a Minor Arc with the centre of a circle is always less than $180\degree$ or $\pi$ radians. In simple words, when a circle is divided into two arcs, the arc with shorter length is called the Minor Arc.

How do you find a minor arc?

The minor arc $(L)$ can be found out by using the formula
$L = 2\pi r\left( {\frac{{{\theta}}}{{360}}} \right)$
where, $\theta$ is the angle subtended by the arc in degrees.

How do you find the radius of a circle with an arc?

The radius of a Circle can be found out with an Arc by using the following formula:
$r = \frac{L}{{2\pi }}\left( {\frac{{360}}{\theta }} \right)$
where, $\theta$ is the angle subtended by the arc in degrees.

Is 180 degrees a major or minor arc?

The Arc having 180 degrees is neither a Major Arc nor a Minor Arc. The Major Arcs are greater than 180 degrees and the Minor Arcs are less than 180 degrees.

How do you find the arc length of an angle?

The Arc length can be found out by using the formula: $L = 2\pi r\left( {\frac{\theta }{{360}}} \right)$
Where r is the radius and $\theta$ is the angle subtended by the arc in degrees.