Arc Length Formula: Definition and Examples

Content Writer

Arc length is defined as the distance along the part of the circumference of a circle. In other words, arc length is any short or long distance along the curve line.

An arc is a part of a circle's curve or circumference.
The distance between two endpoints along a straight line is called a chord.
It is calculated that the length of the arc is greater than that of the length of a straight line.
The length of an arc can be measured using different formulas depending on the unit of the central angle of the arc.
Measurement of the central angle is often given in radians or degrees.
Arc length can be calculated using the differential integration method.

Read More: Circumference of Circle

Table of Content

Arc Length: Definition
What is Radius?
Arc Length Formula
Arc Length Formula for Radians
Arc Length Example
Things to Remember
Sample Questions

Key Terms: Arc Length, Circumference, Segments, Radius, Radians, Curve, Diameter, Central Angle, Differential Integration Method

Arc Length: Definition

[Click Here for Sample Questions]

Arc Length is defined as the interspace between two points on the circumference of any circle. It is always considered as part of a circumference.

A subtended angle by an arc at any point is known to be an angle formed between two line segments joining that point to the endpoint of the arc.
Arc length for a vector value is calculated using the integral formula.
It is a special case of a parametric equation.
An arc can cover any distance along the direction of the curve.
Curve rectification is the process of calculating the length of an irregualr arc.

Fig. Arc Length

Arc Length

Arc length of a circle of radius R that can subtend an angle of θ is given by the equation Rθ. This formula is used when angles are in radians.

What is Radius?

[Click Here for Sample Questions]

Radius is measured as the distance from the center of any circular object to the outermost boundary. A radius is just not only a dimension of a circle but also applicable for a sphere, hemisphere, and any object or shape having a circular base.

Radius

Radius

Arc Length Formula

[Click Here for Sample Questions]

Arc Length Formula is commonly used to find measure of distance along the curved line making up arc. Arc length is the distance that runs through the curved line of a circle.

Angle subtended at the centre of a full circle is 360 degrees.
Mathematically, the Arc Length Formula in calculated in degrees and radians.
For Radian: Arc Length can be calculated in terms of radians, whose formula is as follows:

L = θ x r where θ is the angle expressed in terms of radian, and r is the radius of the circle.

For Degree: Arc length can be calculated in terms of degree, whose formula is as follows:

L = θ x (π/180) x r where θ is the angle expressed in degree and r = radius of the circle.

Length of an arc can be measured using different formulas depending on the unit of central angle of arc.
Measurement of central angle is often given in radians or degrees.
For a circle, arc length formula is known to be θ times radius of a circle.
Arc Length is represented by L.

Fig. Arc Length Formula

Arc Length Formula

Also Read:

Related Articles
Independent Events in Probability	Multiplication Theory on Probability	Conditional Probability Formula
Three Dimensional Geometry	Important questions on circle	Similarity of Triangles

Arc Length Formula for Radians

[Click Here for Sample Questions]

The arc length of a circle can be measured based on the unit of the centre angle of the arc. The arc length formula in radians is:

Arc Length (L) = θ × r

Here, L = Arc Length, θ = Center angle of the arc in radians, and r = Radius of the circle

Read More: Three Dimensional Geometry

Arc Length Example

[Click Here for Sample Questions]

The various examples of arc length in terms of degrees and radians are as follows:

Arc Length Formula for Radians Solved Example

The example to calculate arc length in terms of radians is as follows:

Consider an arc of a circle cut off by a central angle of 7 radians in a circle with a radius of 6 inches.

In that case: Center angle, θ = 7 radians, radius, r = 6 inches.

Use the arc length formula for radians, L = θ × r = 4 × 6 = 42 inches.

Arc Length Formula for Degree Solved Example

The example to calculate arc length in terms of degree is as follows:

Consider a circle where radius of an arc is 5 cm and the central angle is 40°.

In that case: r = 8 cm and Central angle, θ = 40°

Use the arc length formula for degrees, L = θ x (π/180) x r = 40 x (3.14 / 180) x 5 = 3.48 cm

Things to Remember

Arc length is defined as the distance along part of the circumference of any circle.
A curve having a finite length is called a rectifiable curve.
The arc length formula is commonly used to find the measure of distance along the curved line making up the arc.
Measurement of the central angle is often given in radians or degrees.
For a circle, the arc length formula is θ times a circle's radius.
Arc length in radians can be as arc length = θ x r; here θ is in radian
The formula= θ x (π/180) x r gives Arc length in degrees.
The radius is measured as the distance from the centre of any circular object to the outermost boundary.

Read more:

Related Articles
Semicircle	Tangent to a Circle	Algebra
Angle Between Two Lines	Linear Functions	Trigonometric Identities
Areas Related to Circles	Perimeter and Area of a Circle	Collinear Points

Sample Questions

Ques. How do you find arc length with radius and central angle? (2 Marks)

Ans. Arc length of a circle can be calculated with the radius and central angle using the arc length formula,

Length of an Arc = θ × r, where θ is in radian.
Length of an Arc = θ × (π/180) × r, where θ is in degree.

Ques. How can the arc length of a curve be calculated using different methods? (3 Marks)

Ans. The arc length of an arc of a circle can be calculated using different methods and formulas based on the given data. Some important cases are given below,

Find the arc length with the radius along with the central angle
Find the arc length without the radius
Find the arc length without the central angle

Ques. Calculate arc length of a curve, whose endpoints touch a chord of circle measuring 5 units. Central angle subtended by arc is 2 radians. (3 Marks)

Ans. Chord length = 5 units and Central angle = 2 radians

Central angle/2 = 2/2 = 1

Sin(1) = 0.841

Chord length/ (2 × 0.841) = 5/ 1.682 = 2.973 units = radius

Arc length = radius × central angle = 2.973 × 2 = 5.946 units

Hence, arc length = 5.946 units

Ques. How to find arc length without radius? (3 Marks)

Ans. Consider an example to understand the concept of arc length without radius. Calculate arc length of a curve with sector area 25 square units and central angle as 2 radians.

We have, Sector area = 25 units and Central angle = 2 radians

Sector area × 2 = 25 × 2 = 50

50/central angle = 50/2 = 25

√25 = 5

5 × central angle = 5 × 2

= 10 units

Hence, arc length = 10 units

Ques. Calculate the arc length of a curve with sector area 25 square units and radius as 2 units. (3 Marks)

Ans. Sector area = 25 units and Central angle = 2 units

Sector area × 2 = 25 × 2 = 50

50/radius2 = 50/4 = 12.5 = central angle.

Arc length = radius × central angle = 2 × 12.5

= 25 units

Hence, arc length = 25 units

Ques. What is arc length and what is the arc length formula used for? (4 Marks)

Ans. Arc length is defined as interspace between two points on circumference of any circle. It is always considered them part of a circumference. Subtended angle by an arc at any point is known to be the angle formed between the two line segments joining that point to the endpoint of the arc. While Arc length formula is commonly used to find measure of distance along curved line making up the arc.

Arc length is distance that runs through curved line of a circle. If considered between two endpoints of a circle and a straight line, Arc length is bound to be longer than straight line. Basically, length of an arc can be measured using different formulas depending on unit of central angle of arc. Measurement of central angle is often given in radians or degrees. For a circle, arc length formula is known to be θ times the radius of a circle.

Ques. Calculate the arc length of a curve sector area 60cm² and radius measuring as 2 cm. (3 Marks)

Ans. Sector Area is given as 60cm² and radius is given as 2 cm.

Required Area = 60 * 2 = 120 cm²

Central angle = 120 / 4 = 30 radian

Arc Length is calculated as r * central angle which is 30 * 2 = 60 cm.

Ques. Calculate arc length of a curve, whose endpoints touch a chord of circle measuring 6 units. Central angle subtended by arc is 4 radians. (3 Marks)

Ans. Chord length = 6 units and Central angle = 4 radians

Central angle/2 = 4/2 = 2

Sin(1) = 0.841

Chord length/ (2 × 0.841) = 6/ 1.682 = 3.567 units

Arc length = radius × central angle = 3.567 × 4 = 14.26 units

Hence, arc length = 14.26 units

Ques. Find the arc length of a curve on a circle with a radius of 20 cm and a central angle of 5 radians. (3 marks)

Ans. Central angle (θ) is 5 radians. and radius of the circle is 16cm.

Now, Arc length=θ×r

20 * 5
100 cm

Hence, the arc length is 100 cm.

Ques. Calculate the arc length of a curve sector area 30 cm² and radius measuring as 2 cm. (3 marks)

Ans. Curve Sector Area is 30 cm² and radius is 2 cm.

Now,

sector area×2
30 * 2
60

And,

60 /r²
60/ 2 * 2
15

15 is the central angle (In radian).

Then, Arc length= radius×central angle

15 * 2
30 cm

Ques. Express arc length formula in degrees? (1 mark)

Ans. The arc length formula in degree is calculated as θ x (π/180) x r

where θ is the angle expressed in degree and r = radius of the circle.

For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates

Also Check:

Important Study Guides
NCERT Solutions for Class 6 to 12	Class 10 Maths Syllabus	Class 10 Science Syllabus
NCERT Solutions for Class 10 Maths	Class 10 English Syllabus	Class 10 Exam Pattern
CBSE Class 10 Notes	Class 10 Maths Notes	Class 10 Science Notes
NCERT Solutions for Class 10 Science	CBSE Class 10th Sample Papers	Physics Formula
Important Maths Formulas	Math Formula	Chemistry Formula
Maths Study Material	Physics Study Material	Chemistry Study Material
Biology Study Material	Maths Preparation Tips	Periodic Table
Chemical Reactions	Difference between in chemistry	Difference between in physics
Difference between in biology	Physics Constants	Difference between in maths
Named Reactions	Algebra	Arithmetic
Calculus	Geometry	Probability and Statistics
Number System	Trigonometry	Mensuration

Arc Length Formula: Definition and Examples

Arc Length: Definition

What is Radius?

Arc Length Formula

Arc Length Formula for Radians

Arc Length Example

Arc Length Formula for Radians Solved Example

Arc Length Formula for Degree Solved Example

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

4.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

5.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

Comments

SUBSCRIBE TO OUR NEWS LETTER

Arc Length Formula: Definition and Examples

Arc Length: Definition

What is Radius?

Arc Length Formula

Arc Length Formula for Radians

Arc Length Example

Arc Length Formula for Radians Solved Example

Arc Length Formula for Degree Solved Example

Things to Remember

Sample Questions

CBSE X Related Questions

1.Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

2.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

4.Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

5.Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

4.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

5.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is