Arc Length

Let ‘ P ‘ and ‘ Q ‘ be two points on a circles as shown in the figure. Clearly the circle is divided into two pieces each of which is arc of the circle. It denoted from P to Q in anti – clock wise direction by $\angle$PQ and the arc from Q to P in counter clock wise direction by $\angle$QP Note that the points ‘P’ and ‘Q’ lie on both $\angle$PQ and $\angle$QP.
Arc Length

Length of The Arc

The length of an arc is the length of the fine thread which just covers the arc completely.

Major arc
An arc of a circle is called a major arc if its length is greater than the length of the semi – circle.

Minor arc
An arc of a circle is called a minor arc if its length is less than the length of the semi – circle.

Congruent arcs
Two arcs of a circle are congruent, if either of them can be superposed on the other so as to cover it exactly.

Example

An arc of a circle makes the following angles at the centre of the circle. Identify which of them are major arcs, minor arcs and semi circles.

i. 125 degrees       
ii. 260 degrees.   
iii. 180 degrees    
iv.  50 degrees
v. 310 degrees.

Solution:

(i) Since 125 degrees is < 180 degrees, the arc is a minor arc
(ii) Since 260 degrees is > 180 degrees, the arc is a major arc
(iii) Since, 180 degrees is the angle made by diameter at the center of the circle, it is a semi circle.
(iv) Since 50 degrees is < 180 degrees, the arc is a minor arc
(v) Since 310 degrees is > 180 degrees, the arc is a major arc

Arc Length Formula

1. If ‘ θ ‘ is the angle in degree measure made by arc ‘ a ‘ at the centre of the circle of radius ‘ r ’, then the length of the arc is given by

A = $\frac{\theta}{360}$ $2 \pi$ units
2. If ‘ θ ‘ is the angle in radian measure made by arc ‘ a ‘ at the centre of the circle of radius ‘ r ’, then the length of the arc is given by

A = r θ units

 Arc Length Equation


A - r $\theta$ = 0

How to Find Arc Length

Below you could see examples for finding arc length

Parametric Arc Length

In parametric integral form the arc length of curve is given by

$\int_{a}^{b}\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

Arc Length Integral

Formula for arc length of a curve using integrals

Length of the arc $\int_{0}^{3} ds$ =  , That is definite integral of ds from ‘a’ to ‘b’