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Arc Length of a Circle

1.2 Arc Length of a Circle
 
The image contains a diagram explaining the concept of arc length of a circle. On the left, there is a dark blue circle with the text ‘Arc Length of a Circle’ inside it. To the right, there are two labeled boxes connected to the circle with red arrows. The first box, labeled ‘Arc Length,’ contains the definition: ‘The distance along the curved line making up the arc of a circle.’ The second box, labeled ‘Understanding,’ explains that ‘Arc length is a portion of the circumference of a circle.’ The Pandai logo is at the bottom left of the circle.
 
Type of Arc

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.

Diagram depicting a circle with annotations for minor and major arcs clearly illustrated.

In the diagram, note that the angle subtended at the centre of the circle by the major arc is \((2\pi-\theta)\) radians.
 
Formula for Arc Length
Figure

Diagram illustrating a circle with major and minor sector, highlighting radius, angle, and arc length concepts.
When Angle is in Radians
  • Arc Length \(=r\theta\).
  • Where \(r\) is the radius of the circle and \(\theta\) is the angle is radians.
When Angle is in Degrees
  • Arc Length \(=\dfrac{\theta}{360^\circ}\times 2\pi r\).
  • Where \(r\) is the radius of the circle and \(\theta\) is the angle in degrees.
 
Segment
Figure

Diagram illustrating a chord and its segment, highlighting their geometric relationship in a circular context.
Perimeter of the Segment

The perimeter of the shaded segment \(APB\) of a circle is given:

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

or can be write as in the formula below:

An image of a formula showing the perimeter calculation for a segment, including both degrees and radians.

where \(r\) is the radius of the sector and \(\theta\) is the angle of the sector.
 
Example \(1\)
Question

A diagram illustrating a circle within a circle, showcasing a minor sector with a radius of 13 cm and an angle of 2.7 radians.

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

Find the length of the arc \(AB\).
Solution

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

\(\begin{aligned} &=13\text{ cm} \times 2.7 \text{ rad}\\ &=35.1 \text{ cm}. \end{aligned}\)
 
Example \(2\)
Question

Diagram of a minor sector with a 9 cm arc length and shaded segment.

Given length of \(OA=7\) cm,

Find the perimeter of the shaded segment \(APB\).

[ Use \(\pi=3.142\) ]
Solution

Find value of \(\theta\),

\(\begin{aligned} s&=r\theta \\\\ 9&=(7)\theta \\\\ \theta&=\dfrac{9}{7}\text{ rad} .\end{aligned}\)

Convert \(\theta\) to degrees,

\(\begin{aligned} \theta&=\dfrac{9}{7}\text{ rad}\times\dfrac{180^\circ}{3.142} \\\\ &=73.66^\circ .\end{aligned}\)

Find the length of chord \(AB\),

\(\begin{aligned} AB&=2r \sin \dfrac{\theta}{2} \\\\ &=2(7) \sin \begin{pmatrix} \dfrac{73.66}{2} \end{pmatrix} \\\\ &=14 \times \sin 36.83^\circ \\\\ &=14 \times 0. \, 5994 \\\\ &= 8. \,3916 \text{ cm}. \end{aligned}\)

The perimeter of the shaded segment \(APB\),

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

\(\begin{aligned} &=8. \,3916+9 \\\\ &=17.39 \text{ cm} .\end{aligned}\)
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.

Arc Length of a Circle

1.2 Arc Length of a Circle
 
The image contains a diagram explaining the concept of arc length of a circle. On the left, there is a dark blue circle with the text ‘Arc Length of a Circle’ inside it. To the right, there are two labeled boxes connected to the circle with red arrows. The first box, labeled ‘Arc Length,’ contains the definition: ‘The distance along the curved line making up the arc of a circle.’ The second box, labeled ‘Understanding,’ explains that ‘Arc length is a portion of the circumference of a circle.’ The Pandai logo is at the bottom left of the circle.
 
Type of Arc

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.

Diagram depicting a circle with annotations for minor and major arcs clearly illustrated.

In the diagram, note that the angle subtended at the centre of the circle by the major arc is \((2\pi-\theta)\) radians.
 
Formula for Arc Length
Figure

Diagram illustrating a circle with major and minor sector, highlighting radius, angle, and arc length concepts.
When Angle is in Radians
  • Arc Length \(=r\theta\).
  • Where \(r\) is the radius of the circle and \(\theta\) is the angle is radians.
When Angle is in Degrees
  • Arc Length \(=\dfrac{\theta}{360^\circ}\times 2\pi r\).
  • Where \(r\) is the radius of the circle and \(\theta\) is the angle in degrees.
 
Segment
Figure

Diagram illustrating a chord and its segment, highlighting their geometric relationship in a circular context.
Perimeter of the Segment

The perimeter of the shaded segment \(APB\) of a circle is given:

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

or can be write as in the formula below:

An image of a formula showing the perimeter calculation for a segment, including both degrees and radians.

where \(r\) is the radius of the sector and \(\theta\) is the angle of the sector.
 
Example \(1\)
Question

A diagram illustrating a circle within a circle, showcasing a minor sector with a radius of 13 cm and an angle of 2.7 radians.

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

Find the length of the arc \(AB\).
Solution

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

\(\begin{aligned} &=13\text{ cm} \times 2.7 \text{ rad}\\ &=35.1 \text{ cm}. \end{aligned}\)
 
Example \(2\)
Question

Diagram of a minor sector with a 9 cm arc length and shaded segment.

Given length of \(OA=7\) cm,

Find the perimeter of the shaded segment \(APB\).

[ Use \(\pi=3.142\) ]
Solution

Find value of \(\theta\),

\(\begin{aligned} s&=r\theta \\\\ 9&=(7)\theta \\\\ \theta&=\dfrac{9}{7}\text{ rad} .\end{aligned}\)

Convert \(\theta\) to degrees,

\(\begin{aligned} \theta&=\dfrac{9}{7}\text{ rad}\times\dfrac{180^\circ}{3.142} \\\\ &=73.66^\circ .\end{aligned}\)

Find the length of chord \(AB\),

\(\begin{aligned} AB&=2r \sin \dfrac{\theta}{2} \\\\ &=2(7) \sin \begin{pmatrix} \dfrac{73.66}{2} \end{pmatrix} \\\\ &=14 \times \sin 36.83^\circ \\\\ &=14 \times 0. \, 5994 \\\\ &= 8. \,3916 \text{ cm}. \end{aligned}\)

The perimeter of the shaded segment \(APB\),

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

\(\begin{aligned} &=8. \,3916+9 \\\\ &=17.39 \text{ cm} .\end{aligned}\)
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.
Chapter : Circular Measure
Topic : Arc length of a circle
Form 5 Additional Mathematics
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