Arc Length of A Circle

1.2   Arc Length of A Circle
 
  • If the circumference of a circle is divided into two parts of different lengths, the shorter part is known as the minor arc while the longer part is known as the major arc
 

 

In the diagram, note that the angle subtended at the centre of the circle by the major arc is

\((2\pi-\theta)\) radians.

 
 
 

 
  • If \(s\) is the length of an arc of a circle of radius \(r\), that subtends an angle \(\theta\) radians, at the centre \(O\), then
 
\(\begin{aligned} \ s&=r\theta \ \end{aligned}\)
 

Example:

 
Example
     
   

The diagram shows arc \(AB\) of a circle with centre \(O\).

Find the length of the arc \(AB\).

   
     
 

Based on the question, the length of the arc \(AB\)

\(\begin{aligned} &=13\text{ cm} \times 2.7 \text{ rad}\\ &=35.1 \text{ cm}. \end{aligned}\)

 
     
 
 
 
 
  •  The perimeter of the shaded segment \(APB\) of a circle
     
   \(\\ \ = \text{Length of Chord } AB + \text{ Length of Arc } APB\ \)   
     
 
 
 

Example:

 
Example
     
   Find the perimeter of the shaded segment \(APB\).   
     
  \(\begin{aligned} \angle AOB&=\dfrac{9}{7} \text{ rad} \\\\ &=\dfrac{9}{7} \times \dfrac{180}{3.\,142} \\\\ &=73.66 ^\circ. \end{aligned}\)  
     
  \(\begin{aligned} AB&=2j \sin \dfrac{\theta}{2} \\\\ &=2(7) \sin \begin{pmatrix} \dfrac{73.66}{2} \end{pmatrix} \\\\ &=14 \times \sin 36.83^\circ \\\\ &=14 \times 0. \, 5994 \\\\ &= 8. \,3916 \text{ cm}. \end{aligned}\)  
     
 

The perimeter of the shaded segment \(APB\) 

\(\\ \ = \text{Length of Chord } AB + \text{ Length of Arc } APB \\\\\)

\(\begin{aligned} &=8. \,3916+9 \\\\ &=17.39 \text{ cm}. \end{aligned}\)

 
     

 

Arc Length of A Circle

1.2   Arc Length of A Circle
 
  • If the circumference of a circle is divided into two parts of different lengths, the shorter part is known as the minor arc while the longer part is known as the major arc
 

 

In the diagram, note that the angle subtended at the centre of the circle by the major arc is

\((2\pi-\theta)\) radians.

 
 
 

 
  • If \(s\) is the length of an arc of a circle of radius \(r\), that subtends an angle \(\theta\) radians, at the centre \(O\), then
 
\(\begin{aligned} \ s&=r\theta \ \end{aligned}\)
 

Example:

 
Example
     
   

The diagram shows arc \(AB\) of a circle with centre \(O\).

Find the length of the arc \(AB\).

   
     
 

Based on the question, the length of the arc \(AB\)

\(\begin{aligned} &=13\text{ cm} \times 2.7 \text{ rad}\\ &=35.1 \text{ cm}. \end{aligned}\)

 
     
 
 
 
 
  •  The perimeter of the shaded segment \(APB\) of a circle
     
   \(\\ \ = \text{Length of Chord } AB + \text{ Length of Arc } APB\ \)   
     
 
 
 

Example:

 
Example
     
   Find the perimeter of the shaded segment \(APB\).   
     
  \(\begin{aligned} \angle AOB&=\dfrac{9}{7} \text{ rad} \\\\ &=\dfrac{9}{7} \times \dfrac{180}{3.\,142} \\\\ &=73.66 ^\circ. \end{aligned}\)  
     
  \(\begin{aligned} AB&=2j \sin \dfrac{\theta}{2} \\\\ &=2(7) \sin \begin{pmatrix} \dfrac{73.66}{2} \end{pmatrix} \\\\ &=14 \times \sin 36.83^\circ \\\\ &=14 \times 0. \, 5994 \\\\ &= 8. \,3916 \text{ cm}. \end{aligned}\)  
     
 

The perimeter of the shaded segment \(APB\) 

\(\\ \ = \text{Length of Chord } AB + \text{ Length of Arc } APB \\\\\)

\(\begin{aligned} &=8. \,3916+9 \\\\ &=17.39 \text{ cm}. \end{aligned}\)