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# Arc of a Circle – Explanation & Examples

After the radius and diameter, **another important part of a circle is an arc**. **In this article, we will discuss **what an arc is, find the length of an arc, and measure an arc length in radians. We will also study the minor arc and major arc.

## What is an Arc of a Circle?

**An arc of a circle is any portion of the circumference of a circle. To recall, the circumference of a circle is the perimeter or distance around a circle. Therefore, we can say that the circumference of a circle is the full arc of the circle itself.**

## How to Find the Length of an Arc?

Th*e formula for calculating the arc states that:*

**Arc length = 2πr (θ/360)**

Where r = the radius of the circle,

π = pi = 3.14

θ = the angle (**in degrees**) subtended by an arc at the center of the circle.

360 = the angle of one complete rotation.

From the above illustration, the length of the arc (drawn in red) is the distance from point *A* to point *B.*

*Let’s work out a few example problems about the length of an arc:*

*Example 1*

Given that arc,* AB* subtends an angle of 40 degrees to the center of a circle whose radius is 7 cm. Calculate the length of arc* AB.*

__Solution__

Given r = 7 cm

θ = 40 degrees.

By substitution,

The length of an arc = 2πr(θ/360)

Length = 2 x 3.14 x 7 x 40/360

= 4.884 cm.

*Example 2*

Find the length of an arc of a circle that subtends an angle of 120 degrees to the center of a circle with 24 cm.

__Solution__

The length of an arc = 2πr(θ/360)

= 2 x 3.14 x 24 x 120/360

= 50.24 cm.

*Example 3*

The length of an arc is 35 m. If the radius of the circle is 14 m, find the angle subtended by the arc.

__Solution__

The length of an arc = 2πr(θ/360)

35 m = 2 x 3.14 x 14 x (θ/360)

35 = 87.92θ/360

Multiply both sides by 360 to remove the fraction.

12600 = 87.92θ

Divide both sides by 87.92

θ = 143.3 degrees.

*Example 4*

Find the radius of an arc that is 156 cm in length and subtends an angle of 150 degrees to the circle’s center.

__Solution__

The length of an arc = 2πr(θ/360)

156 cm = 2 x 3.14 x r x 150/360

156 = 2.6167 r

Divide both sides by 2.6167

r = 59.62 cm.

So, the radius of the arc is 59.62 cm.

## How to Find the Arc Length in Radians?

There is a relationship between the angle subtended by an arc in radians and the ratio of the length of the arc to the radius of the circle. In this case,

**θ = (the length of an arc) / (the radius of the circle).**

Therefore, the length of the arc in radians is given by,

**S = r θ**

where, θ = angle subtended by an arc in radians

S = length of the arc.

r = radius of the circle.

One radian is the central angle subtended by an arc length of one radius, i.e., *s = r*

The radian is just another way of measuring the size of an angle. For instance, to convert angles from degrees to radians, multiply the angle (in degrees) by π/180.

Similarly, to convert radians to degrees, multiply the angle (in radians) by 180/π.

*Example 5*

Find the length of an arc whose radius is 10 cm and the angle subtended is 0.349 radians.

__Solution__

Arc length = r θ

= 0.349 x 10

= 3.49 cm.

*Example 6*

Find the length of an arc in radians with a radius of 10 m and an angle of 2.356 radians.

__Solution__

Arc length = r θ

= 10 m x 2.356

= 23.56 m.

*Example 7*

Find the angle subtended by an arc with a length of 10.05 mm and a radius of 8 mm.

__Solution__

Arc length = r θ

10.05 = 8 θ

Divide both sides by 8.

1.2567 = θ

There, the angle subtended by the arc is 1.2567 radians.

*Example 8*

Calculate the radius of a circle whose arc length is 144 yards and arc angle is 3.665 radians.

__Solution__

Arc length = r θ

144 = 3.665r

Divide both sides by 3.665.

144/3.665 = r

r = 39.29 yards.

*Example 9*

Calculate the length of an arc which subtends an angle of 6.283 radians to the center of a circle which has a radius of 28 cm.

__Solution__

Arc length = r θ

= 28 x 6.283

= 175.93 cm

**Minor arc (h3)**

The minor arc is an arc that subtends an angle of less than 180 degrees to the circle’s center. In other words, the minor arc measures less than a semicircle and is represented on the circle by two points. For example, arc** AB** in the circle below is the minor arc.

**Major arc (h3)**

The major arc of a circle is an arc that subtends an angle of more than 180 degrees to the circle’s center. The major arc is greater than the semi-circle and is represented by three points on a circle.

For example, PQR is the major arc of the circle shown below.

*Practice Problems*

*Practice Problems*

- Find the area of the sector of the circle of radius 9 mm. Assume the angle subtended by this arc at the center is 30
^{o}. - City A is due North of city B. The latitudes of city A and city B are 54
^{o}N and 45^{o}N, respectively. What is the North-South distance between the two cities? The radius of the Earth is 6400 km.

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