An arc is the part of the circle between two points. Arcs are generally associated with the central angles they make and their measure is expressed as an angular measure using the measure of the central angle.
Arc length is a linear measure of an arc measured along the circumference of the circle.

As arc is a part of the circle, arc measure is a fraction of the circumference of the circle. Arc length is generally indicated by the letter 'l'.

## Arc Length Definition

Arc length is a linear measure of the arc measured along the circle.
It can be understood, that the arc length is a fraction of the circumference of the circle.
To measure the arc length physically:
1. Place a thin wire or thread carefully along the circle.
2. Mark the end points of the arc on the wire or thread.
3. Stretch the wire/thread and measure the distance between the points marked using a Straight edge.

## Arc Length Formula

The full circle can be viewed as an arc where the two end points coincide. A full circle is described by making a central angle of measure 360º.
As an arc length is a fraction of the circumference of the circle, it is proportional to the measure of its central angle.

If the length of arc $\widetilde{AB}$ = l which has a central angle measuring θº and C the circumference of the circle,
then we can write the proportion as
$\frac{l}{C}=\frac{\theta }{360}$
l = $\frac{\theta }{360}\times C$
C can be substituted either by 'πd' or by '2πr' and write the arc length formula as
l = $\frac{\theta }{360}\times \pi d$
or
l = $\frac{\theta }{360}\times 2\pi r$
For major arcs use the reflex angle as θ.

## How to Find Arc Length?

You can use either of the two formulas given above for finding the arc length. But even in case you just could not recollect the formula, there is no reason to worry. You can follow the steps given below to find the arc length.
Step 1: Remember, that the arc length is proportional to its central angle. So the first step is write the proportion $\frac{l}{C}=\frac{\theta }{360}$, where C is the circumference and θ the central angle.
Step 2: Substitute πd or 2πr for C.
Step 3: Substitute the values given and solve for l.

We have the formula to find the arc length given the radius and the central angle.
Reversely the radius of the circle can be found if the arc length and the central angle are given, using the formula,
r = $\frac{180l}{\pi \theta}$ when the central angle θ is given in degrees.

The radian is another way of measuring an angle.

Definition: One radian is the measure of the central angle θ that intercepts an arc equal to the length r the radius of the circle.

### Solved Example

Question: Find the radius of the central if a central angle measuring 120º cuts off an arc measuring 6π cms.
Solution:

Using the above formula,

r = $\frac{180l}{\pi \theta }$          where l = 6π and θ =120º

radius of the circle r = $\frac{180\times 6\pi }{\pi \times 120}$  =  9 cm.

## Arc Length Problems

Let us solve few problems using the formula or steps

### Solved Examples

Question 1: Find the lengths or major and minor arcs $\widetilde{AB}$ correct to the tenth of a cm. The minor arc $\widetilde{AB}$ has a central angle which measures 80º.

Solution:

Radius of the circle r = 5 cm

Using the formula, l = $\frac{\theta }{360}$ $\times 2\pi r$

Length of minor arc $\widetilde{AB}$

= $\frac{80}{360}$ $\times 2\times \pi \times 5$

= $\frac{20 \pi}{9}$ = 7.0 cm

For the major arc we need to use the reflex angle.

Thus θ = 360 - 80 = 280º

Length of major arc $\widetilde{ACB}$

= $\frac{280}{360}$ $\times 2\times \pi \times 5$

= $\frac{70 \pi}{9}$

= 24.4 cm

Question 2: Find the length of a quadrant of a circle whose diameter = 14 inches.
Solution:

The central angle of a quadrant = θ = 90º

If 'l' is the length of the quadrant, we write the proportion as

$\frac{l}{C}$ = $\frac{\theta }{360}$

Step 1: The diameter of the circle is given, hence C can be substituted with πd

$\frac{l}{\pi d}$ = $\frac{\theta }{360}$

Step 2: Substitute d =14 and θ = 90º. $\frac{l}{14\pi }$ = $\frac{90}{360}$

Step 3. l = $\frac{1}{4}$ $\times 14\pi$ Equation simplified and solved for 'l'. ≈ 11 inches.

## Arc Length Practice Problems

### Practice Problems

Question 1: A clock is circular in shape with diameter 12 inches. Find the length of the arc between the Markings 12 and 5 rounded to the hundredth of an inch.
(Answer: 15.71 inches. Hint: Find the angle the hands form when the clock shows 5 º clock.)
Question 2: Find the dotted length in the diagram given below:

(Answer: 13.09 Cm. Find the lengths of two arcs of different circles and add those lengths together.)
Question 3: A square entrance has an arch as showed in the sketch below.  Find the length of the arch nearest to the hundredth of a feet.

(Answer : 19.13 ft.  Hint: You need trigonometry to solve this. The radius of the circle can be found using sine rule and the dotted triangle is an isosceles triangle.)