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Arc Length – Definition, Formula, Examples

Introduction to Arc Length

Arc length refers to the distance between two points along a curve’s section. Arc length formula facilitates in finding the length of a circle’s arc. Furthermore, arc length refers to the distance between two points along a curve’s section.

The process of the determination of the length of an irregular arc is the rectification of a curve. Students can learn more about the arc length and the arc length formula here.

 

arc length formula

Definition and Meaning of Arc Length

Arc length refers to the measure of the distance which exists along a curved line which makes up the arc. Furthermore, it is certainly longer than a straight line distance existing between the endpoints.

The process of the determination of the length of an irregular arc is the rectification of a curve. Furthermore, the emergence of infinitesimal calculus gave rise to a general formula. Moreover, this formula provides solutions to a closed-form nature in some particular cases.

Arc Length Formula and Example(Arc Length Problems)

Length=θ°360°2πr

The arc length formula certainly helps in finding the length of an arc of any circle. Moreover, an arc is an important part of the circumference of a circle.

When an individual works with π, he would desire an exact answer. So, to get an exact answer, one use π. Furthermore, in order to approximate an answer, one must use a rounded form of π, which can be 3.14.

Moreover, r refers to the radius of the circle which certainly happens to be the distance from the centre to the circumference of a circle. Also, the symbol theta, θ, is in usage for the angle degree measures.

An important example is below:

Find the arc length of an arc such that its formation takes place by 60° of a circle with a radius of 8 inches.

Step 1:

Finding the variables.
θ = 60°
r=8

Step 2:

Substitute this into the formula.

Therefore, length= 60°360°2π (8)

Step 3:

One must evaluate for Arc Length in this important step
Length= 16π6
Length= 8π3

In order to get an approximate answer, use 3.14
Length=8(3.14)3
Length =8.37

Answer:

The length comes to be about 8.37 inches.

General Approach Regarding Arc Length Formula

The approximation of a curve in the plane is by connecting a finite number of points on the curve. This takes place by using line segments to produce a polygonal path.

It is pretty straightforward to calculate the length of each linear segment. For example, one can make use of the Pythagorean theorem in Euclidean space.

Moreover, the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance.

Sometimes curve may not already be a polygonal path. Here, one must use a progressively larger number of segments with smaller lengths. This certainly would result in efficient and better approximations.

The lengths of the successive approximations will not decrease. Besides, they may keep increasing indefinitely. However, for smooth curves, they will certainly tend to a finite limit. This takes place as the lengths of the segments become arbitrarily small.

There are some curves, for which there is the smallest number L. This number is an upper bound on the length of any polygonal approximation. Furthermore, these curves are known as rectifiable. Most noteworthy, the number L can certainly be defined as the arc length.

Solved Question for You

Q1. Which of the following statements is not true when it comes to arc length?

A. It refers to the distance between two points along a curve’s section
B. It refers to the measure of the distance which exists along a curved line which makes up the arc
C. Arc length formula certainly helps in finding the length of an arc of any square
D. It is certainly longer than a straight line distance existing between the endpoints

A1. The correct option is C., which is “arc length formula certainly helps in finding the length of an arc of any square.” The correct statement is “arc length formula certainly helps in finding the length of an arc of any circle.”

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