Solving The Arc Length Formula [Some Related Examples]

In mathematics, we can describe an arc as the part of the portion of a curve. We usually discuss the concept of the curve when we talk about the two-dimensional figure, circle. In order words, an arc length formula can be described as the total length covered by the arc along the circumference of a circle of any other curve.

The formula which is used to calculate the total length covered by the arc along the circumference of a circle is called the arc length formula. In this short article, we will discuss the arc length formula and different methods through which the arc length can be calculated. We will also solve some examples related to the topic so that it becomes clearer to you.

What Do You Mean by Arc Length?

Let us suppose that two points extend from the radius of a circle and touch its circumference to form an angle. It forms an arc on the circumference of the circle. The length that is covered by this arc is known as the arc length. For example, we have a circle with the radius r. Two straight lines extend from the radius of the circle and touch the circumference of the circle to form an angle. Let the two points that have been obtained by ‘O’ and ‘P’. Now, OP will be the arc length.

How to Calculate the Arc Length Formula?

The calculation of the arc length can be done using different methods. Which method will be suitable will be dependent on the unit of the central angle that the arc has. We can measure the central angle of the arc either in radians or in degrees. The arc length formula of a circle is θ multiplied by the radius of the circle.

  • The arc length formula when the measure of the central angle of the arc is in radians is = θ * r, where r is the radius of the circle and θ is the measure of the central angle in radian.
  • The arc length formula when the measure of the central angle of the arc is in degrees is = θ * π/180 * r, where r is the radius of the circle and θ is the measure of the central angle in degrees.
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Some Solved Examples Related to the Arc Length Formula

Example 1: What is the length of an arc if the radius of the circle is 12 units and the arc subtends 60 degrees at the center?

Solution: We have learned earlier that,

Length of an arc = θ * π/180 * r

Therefore, length of an arc = 60 * π/180 * 12 = 12.56 units.

 

Example 2: An arc cuts off by a central angle of 3 radians in a circle. The radius of the circle is 12 inches. Find the length of the arc.

Solution: We have learnt that,

Length of an arc = θ * r = 3 * 12 = 36 inches.

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