Arc

Introduction to the Arc

The arc is an important part of geometry and helps us understand the shape and measurement of circles. An arc is only part of the outer edge of a circle. If you imagine a circle in the form of a round pizza, an arc is like one curved slice of the edge.

We can measure an arc in two ways: in degrees or in radians. The formula for arc length helps us find the real distance of the curved part of the circle. Arcs are closely related to other circle parts, such as chords and sectors (circular pizza-like slices).

In this article, we will learn about what an arc is, the circle arc definition, how to calculate the arc length, the difference between the arc, chord, and sector, and how to measure arcs in both degrees and radians. Step-by-step examples will make it easy to understand and use these concepts in solving problems.

Table of Contents

• Definition of Arc
• Arc of a Circle
• Symbol of an Arc
• Measurement of an Arc
• Arc Length Formula
• Solved Examples
• FAQs on Arc

Definition of an Arc

An arc is part of the outer edge of a circle. It is a curved line that combines two points on the circle. Think of a circle like a round ring. If you choose two points on the edge and follow the curve between them, the curved part is called an arc.

Arcs can be small or large, depending on how many circles they cover. The length of the arc is called arc length, and we can measure it by using degrees or radians. Arcs are different from the chords, which are straight lines connecting two points on the circle.

For example, if you cut a small slice ​​of pizza, the curved edge of the slice is like an arc of a circle.

Arc of a circle

An arc of a circle is a part of a circle's edge or circumference. Imagine you have a pizza, and you take a slice; the curved edge of that slice is like an arc. A straight line that joins the 2 ends of the arc is called a chord, and if the arc covers exactly half of the circle, it is called a semicircular arc.

Symbol of an Arc

In geometry, an arc is shown by a small curved line above the letters of the endpoints.

• For example, if the endpoints are A and B, the arc is written as:

AB^ and read as “arc AB”

• The order of points does not matter, so AB^=BA^

Measurement of an Arc

An arc can be measured in 2 ways:

• Angle of the arc

• Length of the arc

The length of the arc is the difference along the curved part of the circle. We measure it in units like centimetres or metres. To show arc length, we use the letter L before the arc name. 

For example:

LAB^=7cm

It is read as “The length of the arc AB is 7 cm.”

Angle of the Arc

The angle formed by the arc at the centre of the circle is called the angle of the arc or central angle.

• It is written as:

mAB^

• Where A and B are the endpoints of the arc.

• Using the arc length formula, we can calculate either the arc length or the angle of the arc.

1. When the angle is in degrees:

Arclength=C×(θ​360°)

θ​=(ArclengthC×360°)

Where:

• C = Circumference of the circle 

• θ = Central angle in degrees

2. When the angle is in radians:

Arclength=r×θ

Where:

• r = Radius of the circle.

• θ = Central angle in radians.
  
  

Arc Length Formula

The arc length is the distance along the curved edge of a circle between 2 points. To find it, we use the arc length formula. The angle made by the arc at the centre of the circle is called the central angle. This angle is very important in calculating arc length.

1. When the angle is in degrees

If the central angle (θ) is given in degrees, then the formula for arc length is:

Arclength=2π×(θ360)

Where:

• θ​ = Central angle in degrees 

• r  = Radius of the circle

Since the circumference of a circle is

C=2πr

We can also write:

Arclength=C×(θ360)

2. When the angle is in radians

IF the central angle (θ) is given in radians, then the formula for arc length is 

Arclength=r×θ

 

Solved Examples

Example 1: A circle has radius r = 7cm. The arc makes an angle of 60° at the centre. Find the arc length.

Solution:

Arclength=2π×(θ360)

Arclength=2π×(60360)×7

Arclength=14π×(16)

Arclength=(14π6)=(7π3)cm

Example 2: In a circle, the radius is 10 cm. The central angle is 90°. Find the arc length.

Solution:

Arclength=2π×(θ360)

Arclength=2π×(90360)×10

Arclength=20π×(14)

Arclength=5πcm

Example 3: A circle has a radius of 12 cm. The angle made by the arc at the centre is 2 radians. Find the arc length.

Solution: Formula in radians:

Arc length = r × θ

                   = 12 × 2

                   = 24 cm

Example 4: A circle has a radius of 5 cm. The arc makes an angle of π3 radians at the centre. Find the arc length.

Solution: Formula in radians:

Arc length = r × θ

                   = 5 × π3

                   = 5π3cm

 

FAQs on Arc

1. What is the arc of a circle?

The arc of a circle is the curved part of the circle's edge between 2 points. It's like a small piece of the circle's boundary.

2. What are the chord and arc of a circle?

• Arc: The curved part of the circle.

• Chord: A straight line that joins the 2 ends of an arc.

• Example: In a pizza slice, the curved edge is the arc, and the straight edge is the chord.

3. Is the arc of a circle 360°?

Yes, if we take the whole circle as an arc, then its angle is 360°. Smaller arcs are always less than 360°.

4. What is an arc shape?

An arc shape looks like a curved line or a bow. It is not straight like a chord but bent like a piece of the circle's edge.

5. How do you find the arc of a circle?

To find the length of an arc, we use the arc length formula

• If the angle is in degrees:

Arclength=(θ360°)×2πr

• If the angle is in radians:

Arclength=r×θ

• Here, r is the radius and θ​ is the angle at the centre.