Arc Length Formula

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What is the Arc Length Formula?

To find the length of an arc of a circle, let us understand the arc length formula. An arc is a component of a circle's circumference.

Again, if we want an exact answer when working with π, we use π. We substitute a rounded form of π, such as 3.14, if we want to approximate a response. Also, r refers to the radius of the circle, which is the distance from the center to the circumference of a circle.


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What is an Arc?

An arc is a connected subset of a separable curve in Euclidean geometry. Depending on whether they are confined or not, arcs of lines are called segments or rays. Two arcs are determined by every pair of distinct points on a circle. If the two points are not directly opposite each other, an angle at the center of the circle that is less than π radians (180 degrees) will be subtended by one of these arcs, the minor arc, and an angle greater than π radians by the other arc, the major arc.


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What is Arc Length?

Now, let us understand what is arc length. Arc length is the distance along a segment of a curve between two points. Rectification of a curve is often called evaluating the length of an irregular arc section. The advent of infinitesimal calculus led to a general arc angle formula which, in some instances, provides closed-form solutions.


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Finding Arc Length

Now, let us find out how to find the arc length under different conditions. For finding arc length, there are different arc angle formula for different conditions.

  • Arc Length Formula Radians

If θ is given in radians, 

S = θ × r


  • Arc Length Formula Degrees

If θ is given in degrees

S = 2πr(θ/360)


  • Arc Length Formula Integral Form

Integral form

\[S = \int_{a}^{b} \sqrt{1 + (\frac{dy}{dx})^{2} dx}\]

 

Where, s: arc length of the circle,

r: radius of the circle,

θ: central angle of a circle.


Conclusion

The arc formula is used to find the length of an arc in the circle. And as seen above, there are different formulas under different conditions. So, while we calculate the arc length we have to focus on the given conditions.

FAQ (Frequently Asked Questions)

1. What is the Arc Formula for a Circle?

Solution: Arc length = 2πr(θ/360)

2. Can the Length of Arcs be Negative?

Solution: A curve's arc length can not be negative, just as the distance can not be negative between two points.

3. What are the Two Ways to Make an Arc Begin?

Solution: Scratching and tapping are the two ways to begin an arc.