Arc of a Circle

Arc of a Circle is a part of the circumference of a circle or we can also say the Arc of a Circle is some percentage of the circle’s circumference. As we know, a circle is defined as a two-dimensional geometrical object where all the points are equidistant from the center and the distance measured around the circle is known as a circumference and some portion of the circumference taken at a time is known as the Arc of a Circle.

In this article, we will learn the Arc of a Circle in detail, including its definition, types, and arc length formula. Other than that we will also discuss the angle subtended by an arc and the theorem related to this angle as well.

What is the Arc of a Circle?

A better method to describe arc length is the distance around the circumference of any circle or curve (arc). Any distance along the curved route that makes up the arc is measured by its length. An arc is a section of a curve or the outside of a circle. Each of them is shaped like a curve. Any chord between the endpoints of an arc is longer than any distance in a straight line. Any section of a circle’s circumference is considered an arc. Remember that the circumference of a circle is its perimeter or distance. As a result, we may state that the circumference of a circle equals the circle’s whole arc.

Arc of a Circle Definition

The arc of a circle is referred to as a segment or part of its circumference.

A straight line can be made by linking the two ends of an arc to form a circle’s chord. Any arc that is precisely half the diameter of a circle is said to be semicircular.

How to Make an Arc of a Circle?

To make arc, we can use following steps:

• Pick any three distinct points that are not collinear.
• Draw straight lines connecting each pair of points.
• Construct the perpendicular bisectors of both lines and locate their point of intersection. This is the circumcircle’s center.
• Choose the distance between the center and any of the three points as the radius.
• Using the chosen radius and the center point, draw a circle.

Exact steps are illustrated in the following illustration:

Types of Arcs

A circle is divided into two sections by an arc, as you must have observed.

• Minor Arc
• Major Arc
• Semi Circle

Minor arc of a circle

A circle’s minor arc is essentially less than half of the circle’s overall arc. Blue color curve in the following figure is the minor arc in the circle.

Major Arc of a Circle

The major arc of the main circle is the arc that extends more than half of the circle. Red color curve in the following figure is the major arc in the circle.

Semi Circle

A semicircle is described in geometry as a half circle generated by cutting a circle in half. A line passing through the centre and touching the two extremities of the circle creates it. This line is known as the circle’s diameter.

Arc of the Circle Formula

The formula shown below can be used to determine an arc’s length.

Arc Length of the Circle = 2πr(θ/360°)

Where,

• r denotes the radius of the circle,
• 360° the angle of one full revolution, and
• θ which is the centre angle of the arc.
• π (Pi) has a value of 3.14.

Simplifying this formula further we get,

Arc Length of the Circle = (θ/360°) 2πr = (θ/180°) πr

Read More,

• Radius of a Circle
• Diameter of a Circle
• Circumference Formula

How to Find Length of Arc of a Circle?

Here’s a step-by-step explanation of how to find the length of an arc of a circle, using an example.

Assume we have a circle with a radius of 10 units and a centre angle of 120°. The length of the arc subtended by this angle is what we’re looking for.

Step 1: Check the Given values

• Radius (r) = 10 units
• Central angle (θ) = 120°

Step 2: Calculate Arc Length

Using the formula for arc length:
Arc Length = (θ/360°) × 2πr

Step 2.1: Calculate Fraction of Circle’s Circumference

θ / 360° = 120°/360° = 1/3

Step 2.2: Calculate Arc Length

Arc Length = (1/3) × 2π × 10

⇒ Arc Length = (2/3)π × 10

⇒ Arc Length = (20/3)π units

Step 3: Finalize the Result

If you want a numerical estimate, the value of π (pi) is about 3.14.

Thus, Arc Length ≈ (20/3) × 3.14 ≈ 20.93 units

The arc subtended by a central angle of 120° in a circle with a radius of 10 units is roughly 20.93 units long.

How to Find the Arc Length in Radians?

The angle that an arc occupies in radians and the proportion of the arc’s length to the circle’s radius is related. In this instance.

θ = (Length of an Arc)/(Radius of the Circle)

OR

S = r θ

Where,

• θ is the angle in radians that an arc occupies,
• S is the angle’s length, and
• r is the radius of the given circle.

Note:

• For θ = 1 radian, or s = r, is the center angle that a radius-length arc subtends.
• The radian is merely another unit of measurement for angles. For instance, multiply the angle (in degrees) by π/180 to convert angles from degrees to radians.
• The angle (in radians) is multiplied by 180/π to convert from radians to degrees.

Angle Subtended by Arc at Center

The angle subtended by an arc at the centre of a circle is the angular measure created by two radii commencing from the centre and continuing to the arc’s ends. It is the basic connection between the central angle and the appropriate arc length. This angle, given in radians or degrees, controls the length of the arc, with a direct ratio to the radius.

Theorem of Angle Subtended by Arc at Center

Given: An arc PQ of a circle is given, bounded by angles ∠POQ at the centre O and ∠PAQ at a point A on the remaining half of the circle.

To prove : ∠POQ=2∠PAQ

Construction : Join the line that AO extended to B.

Proof:

∠BOQ=∠OAQ+∠AQO  . . . (1)

Also, in △ OAQ,

OA=OQ [The radius of a circle]

Therefore,

∠OAQ=∠OQA      [Angles opposing equal sides have the same value]

∠BOQ=2∠OAQ . . . (2)

Similarly, BOP=2∠OAP    . . . (3)

Adding 2 & 3, we get,

∠BOP+∠BOQ=2(∠OAP+∠OAQ)

∠POQ=2∠PAQ . . . (4)

For instance 3, where PQ is the main arc, equation 4 is substituted by Reflex angle, ∠POQ=2∠PAQ.

Read More,

• Circumfernce of Circle
• Sector of a Circle
• Chord of a Circle

Solved Examples on Arc of a circle

Example 1: Using 48 cm, determine the length of an arc of a circle that forms a 160° angle with the circle’s center.

Solution:

Given:

The value of π = 3.14

The value of θ = 160

The value of r = 48

The length of an arc = 2πr(θ/360)

= 2×3.14×48×160/360

= 133.97 cm.

Example 2: The radius of the circle is 18 units and the arc subtends 85° at the center. How long is the arc, measured in terms of circumference?

Solution:

We know that,

Circle Circumference = 2πr

C = 2π × 18

= 36π

Arc length = (θ/360) × C

= (85°/360°)36π

= (85°/360°)36×3.14

= 26.69 units

Arc length = 26.69 units

Example 3: In an arc with a radius of 20 cm and an angle subtended of 0.456 radians, determine its length.

Solution:

Given:

Radius (r) = 20

Radians (θ) = 0.456

Arc length = r θ

= 0.456×20

= 9.12 cm.

Example 4: A circle having a radius of 6 mm and a length of 15.06 mm should have an angle subtended by it.

Solution:

Given:

Arc length = 15.06 mm

Radius(r) = 6 mm

Arc length = r θ

15.06 = 6 θ

Divide both sides by 6.

2.51 = θ

Thus, the arc there subtends an angle of 2.51 radians.

Practice Problems on Arc of a Circle

Problem 1: Find the length of the arc of a circle with a central angle of 45° and a radius of 8 inches.

Problem 2: If the measure of an arc in a circle is 120°, and the radius is 6 centimeters, what is the length of the arc?

Problem 3: Given a circle with a radius of 10 meters, find the measure of the central angle if the length of the arc is 15 meters.

Problem 4: Calculate the length of an arc in a circle with a radius of 5 inches if the central angle is 60°.

Problem 5: A sector of a circle has a central angle of 90° and a radius of 12 centimeters. Find the length of the arc and the area of the sector.

Arc of a Circle – FAQs

1. What is the circle’s arc?

The circumference of a circle’s arc, which lies between any two points on it, is measured in length. i.e., Any portion of a circle’s circumference is an arc. The arc’s angle is the angle generated by two line segments connecting a point to the endpoints of an arc at any given place.

2. What is the formula for the arc length of a circle?

A circle’s radius (r) and central angle (θ) are two variables used in the calculation for arc length. L stands for it, and the formula is

the equation L = rθ (π/180) if θ is given in degrees

if θ is in radians, then L = rθ is the formula.

3. How to Determine the Length of an Arc Using Radians?

The arc of a circle formula may be used to get the arc length when the central angle is specified in radians, which is given as L  = θ × r, when θ is in radians.

Where,

• L is the Arc Length,
• θ is the Center angle of the arc, and
• r is the circle radius.

4. What is the Angle in the Central?

The central angle is the angle that the arc at the center of the circle subtends.

5. What Angle is Inscribed?

An inscribed angle is the angle that the arc occupies at any point along the circle’s circumference.

6. How Can You Calculate an Arc’s Length Without the Radius?

The circle’s radius and center angle are unquestionably necessary in order to determine the arc’s length. However, if the radius is omitted, the sector area or chord length may have been provided instead. Apply the arc length formula after using the following calculations to find the radius.

• Sector area = (θ/360) × πr², if θ is in degrees (or) (1/2) r²θ
• Chord length = 2r sin (θ/2)