Arc Length: Definition, Formula, Solved Examples & More

What is an arc and what are the various methods to find its length? Check all formulas along with solved examples to understand the application of all the formulas to calculate arc length.

Created On: Jun 29, 2021 20:22 IST
Arc Length Formula
Arc Length Formula

What is Arc Length?

Arc length is the distance between two points along a section of a curve.

Or

Arc length can also be defined as the distance along the part of the circumference of a circle.

Arc Length Formula

Arc length can be calculated with the help of the central angle of that arc and the radius of the circle to which that arc belongs. The unit of the central angle can be in degrees or radians. So, depending upon the unit of the central angle the length of the corresponding arc can be found out using different formulas as mentioned below:

When the unit of central angle is in radians then

Arc length, s = θ × r

When the unit of central angle is in degrees then

Arc length, s = 2πr (θ/360o) = πr (θ/180o)

Where,

s = Length of arc

θ = Central Arc Length of Arc

r = Radius of the circle            

Arc length of a curve in integral form:

The arc length of a curve is given as

Where,

f′(x) = First integral of the given curve whose arc length is to be calculated

b = It is the upper limit of our definite integral that is the rightmost point of the given arc

a = It is the lower limit of our definite integral that is the leftmost point of the given arc

1. Let the angle subtended by an arc at the centre of the circle be 120o. If the circumference of the circle is 54 cm then find the arc length.

Solutions:

Given,

Central angle, θ = 120o

Circumference of circle, 2πr = 54 cm

Thus, arc length is

Thus the arc length comes to be 18 cm.

2. Find the central angle subtended by an arc whose length is 1/4 of the circumference of the corresponding circle.

Solution:

We know that circumference of the circle, C = 2πr

And Arc length, s = 2πr (θ/360o)

According to the question,

Hence, the central angle = 90o.

3.Find the arc length of the curve y = 4 between points x = 0 and x = 4.

Solution:

Given, f(x) = 4

Therefore, the required arc length is 4 units.

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