Arc – Definition with Examples

What is Arc in Math?

In mathematics, an arc is defined as a portion of the boundary of a circle or a curve. It can also be referred to as an open curve.

The boundary of a circle is the perimeter or the distance around a circle, also known as the circumference. So, the arc is the distance between any two points traced along its circumference. 

Let us understand this with the help of an example:

An arc

In this figure, the distance between points A and B is an arc traced along the circumference of the circle. You can call it arc AB. An arc is denoted by the symbol “⌢”. So, the arc AB can be written as $\widehat{AB}$. You can also write it as $\widehat{BA}$. The order of the points does not matter.

How to Make an Arc

To make an arc, you either need a chord or a central angle. 

A chord is a line segment that joins any two points on a circle. 

A chord

A central angle is an angle between any two radii of the circle. For instance, the central angle in the diagram between the radii QA and QB, as shown below, is 60°.

A central angle of 60 degrees

Types of Arcs

You must have noticed that an arc divides a circle into two parts.

One has a shorter distance between the two endpoints (minor arc), while the other has a longer distance (major arc).

Minor arc and major arc

Unless specified, an arc will always be considered a minor arc. To specify the major arc, you can take a third point on the arc of the circle and use three letters in the name.

Central angle, minor arc, and major arc

In the diagram above, $\widehat{AB}$ is the minor arc, and $\widehat{ADB}$ is the major arc.

Semicircle Arc

A semicircle is an arc that has its endpoints on the diameter of a circle.

Semicircular arcs

How to Find the Arc Length?

You can calculate the length of an arc by using the formula given below.

Arc length of a circle = $\frac{y}{360}$  ✕ $2 π r$

Where r = the radius of the circle

y = the angle (in degrees) subtended by an arc at the center of the circle

360 = the angle of one complete rotation.

The value of $π$ (pi) = 3.14

Solved Examples

Example 1: Calculate the length of an arc that subtends an angle of 60 degrees at the center of a circle with a radius of 5 cm.

Solution: We know the arc length formula is $\frac{y}{360} ✕ $2 π r$

In this example, 

y = 60 and r = 5

Substituting these values in the example, we get

Arc length = $\frac{60}{360}$ ✕ 2 ✕ 3.14 ✕ 5 = 5.23 cm

Example 2: Calculate the length of the arc that subtends an angle of 40 degrees at the center of a circle with a radius of 6 cm.

Solution: We know the arc length formula is $\frac{y}{360}$ ✕ $2 π r$ 

In this example, 

y = 40 and r = 6

Substituting these values in the example, we get

Arc length = $\frac{40}{360}$ ✕ 2 ✕ 3.14 ✕ 6 = 4.186 cm

Example 3: Identify the major arc in this circle.

A circle with ADC as major arcSolution:

We know that a major arc is a long distance between two endpoints. So here, $\widehat{ADC}$ is the major arc, and $\widehat{ABC}$ is the minor arc.

Practice Problems

Arc - Definition with Examples

Attend this quiz & Test your knowledge.

1Calculate the length of the arc that subtends an angle of 120 degrees at the center of a circle with a radius of 10 cm.

20.93 cm
10.93 cm
14.56 cm
30.46 cm
CorrectIncorrect
Correct answer is: 20.93 cm
Arc length = $\frac{y}{360}$ ✕ $2 π r$
Substituting y = 120 and r = 10 cm, we get
Arc length = $\frac{40}{360}$ ✕ 2 ✕ 3.14 ✕ 10 = 20.93 cm

2Identify the minor arc in the figure given below.

$\widehat{AB}$
$\widehat{BD}$
$\widehat{DA}$
$\widehat{ADB}$
CorrectIncorrect
Correct answer is: $\widehat{AB}$
$\widehat{AB}$ is the minor arc in this diagram.
Arc – Definition with Examples

3An arc on the circle is drawn so that it covers a quarter of the circumference. What will be the value of the central angle in this case?

90°
60°
45°
30°
CorrectIncorrect
Correct answer is: 90°
For an arc that spans a quarter of the circle, the radii drawn from its endpoints will be perpendicular to each other.
Arc – Definition with Examples

4Which of these central angles will create the greatest arc?

81°
76°
45°
99°
CorrectIncorrect
Correct answer is: 99°
The central angle is formed by joining the endpoints of the arc with the center of the circle. Hence, the greater the central angle, the greater will be the length of the arc formed.

Frequently Asked Questions

A semicircle is an arc that has its endpoints on the diameter of the circle.

Two Semicircle arcs

Here, both $\widehat{AB}$ and $\widehat{ACB}$ are semicircle arcs.

A central angle is an angle that an arc subtends at the center.

No. An arc is always an open curve.