An Arc is a Continuous Portion of a Circle

Arc Length Formula: Definition, Types, and Examples

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Arc Length Formula: A continuous part of a curve or a circle's circumference is called an arc. Arc length is defined as the distance along the circumference of any circle or any curve or arc. The curved portion of allobjects is mathematically called an arc. If two points are chosen on a circle, they divide the circle into one major arc and one minor arc or two semi-circles.

The length of an arc is longer than any line distance between its endpoints (a chord). In this article, we will study the definition of an arc, the length of an arc formula in different types, and solve some example problems.

What is an Arc?

A continuous piece of a circle is called an arc of a circle.

Consider a circle \(C(O,\,r)\). Let \({X_1},\,{X_2},\,{X_3},\,{X_4},\,{X_5},\,{X_6}\) be the points on the circle. Then, the pieces \({X_1}{X_2},\,{X_3}{X_4},\,{X_5}{X_6},\,{X_1}{X_3}\) etc. are all arcs of the circle \(C(O,\,r)\).

Let \(X\) and \(Y\) be two points on a circle \(C(O,\,r)\). The circle is divided into two pieces, each of which is an arc. We denote the arc from \(X\) to \(Y\) in the counterclockwise direction by \(XY\) and the arc from \(Y\) to \(X\) in the counterclockwise direction by \(YX\). Note that the points \(X\) and \(Y\) lie on both \(XY\) and \(YX\).

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Arc Length of a Circle

The length of an arc \(XY\) is the length of the fine thread that covers the arc completely. We denote the length of an arc by \(l\left( {XY} \right)\).

From the above discussion, we have, for any two points \(X\) and \(Y\) on a circle either \(l\left( {XY} \right) < l\left( {YX} \right)\)or \(l(XY) = l(YX)\) or \(l(XY) > l(YX)\)

If \(l(XY) < l(YX)\), then \(XY\) is called the minor arc, and \(YX\) is known as the major arc. Thus, the arc \(XY\) will be a minor arc or a major arc according \(l(XY) < l(YX)\) or \(l(XY) > l(YX)\)

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Central Angle

Let \(C(O,\,r)\) be any circle. Then any angle whose vertex is \(O\) is called the central angle. In the below figure \(\angle XOY\) is a central angle of the circle \(C(O,\,r)\).

The definition of minor and major arcs of a circle by using the concept of central angles are given below:

Minor Arc

A minor arc of a circle is the collection of all the points that lie on and inside a central angle. In other words, a minor arc of a circle is a part of the circle intercepted by a central angle, including the two points of intersection.

Major Arc

A major arc of a circle is the collection of circle points that lie on or outside a central angle. In the below figure, \(XY\) is a minor arc, and \(YX\) is a major arc of the circle.

It is evident from the above discussion that the length of an arc is closely related to the central angle determining the arc. The larger the central angle, the larger are going to be the minor arc.

Length of the Arc Formula

The length of an arc can be calculated using different methods and formulas using the given data. They are:

Arc length formula using radius and central angle
Arc length formula using central angle and sector area
Arc length formula using the radius and sector area
Arc length formula using integrals

Arc Length of a Circle Formula Using Radius and Central Angle in Radians

The length of an arc when the radius and central angle in radians are given by:

\(l = r\theta \)

Where, \(l = \) length of an arc,
\(r = \) radius,
\(\theta = \) central angle in radians.

Arc Length Formula Using Radius and Central Angle in Degrees

The length of an arc when the radius and central angle in degrees is given by:

\(l = 2\pi r\left( {\frac{\theta }{{{{360}^{\rm{o}}}}}} \right)\) or \(l = \frac{{\theta \pi r}}{{{{180}^{\rm{o}}}}}\)

Where, \(l = \) length of an arc,
\(r = \) radius,
\(\theta = \) central angle in degrees.

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Arc Length Formula Using Central Angle and Sector Area

Let \(OAPB\) be a sector of a circle with centre \(O\) and the radius \(r\). Let the degree measure of \(\angle AOB\) be \(\theta \).

We know that the area of the circle is \(\pi {r^2}\).

We can consider this circular region a sector forming an angle of \({360^{\rm{o}}}\) at the centre \(O\). Now by applying the unitary method, we can arrive at the area of the sector \(OAPB\) as follows.

When the degree measure of the angle at the centre is \({360^{\rm{o}}}\), the area of the sector is \(\pi {r^2}\).

So, when the degree measure of the angle at the centre is \(1\), the area of the sector \( = \frac{{\pi {r^2}}}{{{{360}^{\rm{o}}}}}\).

Therefore, when the degree measure of the angle at the centre is \(\theta \), the area of the sector is \(\frac{{\pi {r^2}}}{{{{360}^{\rm{o}}}}} \times \theta = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2}\)

Thus we obtain the following formula for the area of a sector of a circle:

Area of a sector of angle \(\theta = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2}\)

Where \(r\) is the radius of the circle and \(\theta \) is the angle of the sector in degrees.

Again by applying the unitary method and taking the whole length of the circle of angle \({{{360}^{\rm{o}}}}\) as \(2\pi r\), we obtain the required length of the arc \(APB\) as \(\frac{\theta }{{{{360}^{\rm{o}}}}} \times 2\pi r\)

So, the length of an arc of a sector of angle \(\theta = \frac{\theta }{{{{360}^{\rm{o}}}}} \times 2\pi r\)

Arc Length Formula Using Integrals

The formula to find the arc length of a function in integral form is given by:

\(l = \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}dx} } \)

Where \(a\) and \(b\) are the intervals.

\(\frac{{dy}}{{dx}}\)is the derivative of the given function.

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Solved Examples – Arc Length Formula

Q.1. Calculate the length of an arc if the radius of an arc is \(5\,{\rm{cm}}\) and the central angle is \({45^{\rm{o}}}\). (Take \(\pi = 3.14\))
Ans:
Given: Radius \(r = 5\;{\rm{cm}}\)
Central angle \(\theta = {45^{\rm{o}}}\)
We know that arc length \(l = \frac{\theta }{{{{360}^{\rm{o}}}}} \times 2\pi r\)
\( \Rightarrow l = \frac{{45}}{{360}} \times 2 \times \pi \times 5\)
\( \Rightarrow l = 3.925\;{\rm{cm}}\)
Therefore, the length of the arc is \(3.925\;{\rm{cm}}\).

Q.2, In a circle of radius \(21\;{\rm{cm}}\), an arc subtends an angle of \({60^{\rm{o}}}\) at the centre. Find the length of the arc.
Ans:
Given: Radius \(r = 21\;{\rm{cm}}\)
Central angle \(\theta = {60^{\rm{o}}}\)
We know that arc length \(l = \frac{\theta }{{{{360}^{\rm{o}}}}} \times 2\pi r\)
\( \Rightarrow l = \frac{{60}}{{360}} \times 2 \times \pi \times 21\)
\( \Rightarrow l = 7\pi \,{\rm{cm}}\)
Therefore, the length of an arc is \(7\pi \,{\rm{cm}}\).

Q.3. A brooch is made with silver wire in the form of a circle with a diameter of \(35\,{\rm{mm}}\). The wire is also used in making \(5\) diameters which divide the circle into \(10\) equal sectors. Find the total length of the silver wire required and the arc length of each sector.

Ans: It is given that the brooch is made with a silver wire in the form of a circle.
Also, the diameter of the brooch is \(35\,{\rm{mm}}\)
So, the radius of the brooch is \(\frac{{35}}{2}\,{\rm{mm}}\)
Since the silver wire is used in making \(5\) diameters and circles.

So, total wire is used \( = \) length of wire in circle \(+\) wire used in \(5\) diameters
Length of wire \( = \) Circumference of a circle
\( = 2\pi r\)
\( = 2 \times \frac{{22}}{7} \times \frac{{35}}{2}\)
\( = 22 \times 5 = 110\;{\rm{mm}}\)
The silver wire used in \(5\) diameters \( = 5 \times \) diameter of the circle
\( = 5 \times 35 = 175\;{\rm{mm}}\)
Now, total wire is used \( = \) length of wire in circle \( + \) wire used in \(5\) diameters
\( = 110 + 175\)
\( = 285\;{\rm{mm}}\)

Now, we have to find the length of the arc of each sector.
\( = \frac{{2\pi }}{{{\rm{number}}\,{\rm{of}}\,{\rm{sectors}}}} = \frac{{2\pi }}{{10}} = \frac{\pi }{5}\) radians
Also, we know that the length of an arc \( = r\theta \).
Where \(\theta \) is the angle of the sector at the centre in radians.
Therefore, we get the arc length of the sector \( = r \times \frac{\pi }{5}\)
Also, \(r = 17.5\;{\rm{mm}}\) and \(\pi = \frac{{22}}{7}\).
Therefore, the arc length of the sector \( = 17.5 \times \frac{{22}}{{7 \times 5}} = 11\,{\rm{mm}}\)

Q.4. Find the circle radius where a central angle of \({60^{\rm{o}}}\) intercepts an arc length of \(37.4\;{\rm{cm}}\).
Ans: Given: \(l = 37.4\;{\rm{cm}}\) and \(\theta = {60^{\rm{o}}} = \frac{{60\pi }}{{180}}\) radians \( = \frac{\pi }{3}\)
We know that arc length \(l = r\theta \Rightarrow r = \frac{l}{\theta }\)
\( \Rightarrow r = \frac{{37.4 \times 3}}{{\frac{{22}}{7}}}\)
\( \Rightarrow r = \frac{{37.4 \times 3 \times 7}}{{22}}\)
\( \Rightarrow r = 35.7\;{\rm{cm}}\)
Therefore, the radius of the given circle is \(35.7\;{\rm{cm}}\).

Q.5. The minute hand of a watch is \(1.5\;{\rm{cm}}\) long. How far does its tip move in \(40\) minutes? (Use \(\pi = 3.14\))
Ans: In \(60\) minutes, the minute hand of a watch completes one revolution. Therefore, in \(40\) minutes, the minute hand turns through \(\frac{2}{3}\) of a revolution. Therefore \(\theta = \frac{2}{3} \times {360^{\rm{o}}}\) or \(\frac{{4\pi }}{3}\) radians.
Hence the required distance travelled is given by \(l = r\theta = 1.5 \times \frac{{4\pi }}{3}\;{\rm{cm}} = 2\pi {\rm{cm}} = 2 \times 3.14\;{\rm{cm}} = 6.28\;{\rm{cm}}\)
Therefore, the minute hand tip moves \(6.28\;{\rm{cm}}\) in \(40\) minutes.

Summary – Arc of a Circle Formula

In this article, we have learnt the definitions of arc, central angle, length of an arc, major arc, and minor arc. Also, we have derived the formula to find the length of the arc and studied the different formulas to find the length of an arc, and solved some example problems on the same.

FAQs on Arc Angle Formula

Q.1. How do you calculate arc length?
Ans: We find the arc length using the formula \(l = r\theta \)
Where, \(l = \) length of an arc, \(r = \) radius and \(\theta = c\) central angle in radians.

Q.2. What is the formula for arc length in degrees?
Ans: The length of an arc when the radius and central angle in degrees is given by:
\(l = 2\pi r\left( {\frac{\theta }{{{{360}^{\rm{o}}}}}} \right)\) or \(l = \frac{{\theta \pi r}}{{{{180}^{\rm{o}}}}}\)
Where \(l = \) length of an arc, \(r = \) radius, and \(\theta = c\) central angle in degrees.

Q.3. What is the length of an arc in integrals?
Ans: The formula to find the arc length of a function in integral form is given by:
\(l = \int\limits_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}dx} } \)
Where \(a\) and \(b\) are the intervals and \(\frac{{dy}}{{dx}}\)is the derivative of the given function.

Q.4. What are arc length and sector area?
Ans: The length of an arc is the length of the fine thread that covers the arc completely. The amount of space enclosed within the boundary of a sector is called the area of a sector of a circle.

Q.5. What are major arc and minor arc?
Ans: A major arc of a circle is the collection of circle points that lie on or outside a central angle. A minor arc of a circle is the collection of the circle points that lie on and inside a central angle.

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