1.2 Arc Length of a Circle
Arc length is the distance along the curved line making up an arc of a circle. In this topic, students learn the proportional relationship between the angle subtended at the centre and the arc length, deriving the formula \(s = r\theta\), where \(\theta\) must be in radians. Students will also learn how to determine the perimeter of segments by combining arc lengths and chord lengths, and apply these concepts to various practical real-world problems.
Learning Objectives
- Determine the arc length of a circle, the radius, and the angle subtended at the centre of a circle using the formula \(s = r\theta\).
- Determine the perimeter of a segment of a circle using arc length and chord length rules (such as the cosine rule or right-angled trigonometry).
- Solve real-world mathematical problems involving arc lengths, perimeters of shapes, and circular measures.
Anatomy of a Circle Sector and Segment
See how radius \(r\), arc length \(s\), chord, and segment perimeter relate to the central angle \(\theta\) in radians.
Sector \(AOB\)
The sector is bounded by two radii \(OA\) and \(OB\), and the arc \(AB\). Its curved boundary is the arc length \(s = r\theta\).
Segment \(ABC\)
The segment is bounded by the arc \(ABC\) and the chord \(AC\). Its perimeter is the sum of the arc length and the chord length.
Arc length when the angle is in radians
\(s = r\theta\)where \(s\) is arc length, \(r\) is radius, and \(\theta\) is the central angle in radians.
Radius
\(r = \dfrac{s}{\theta}\)Given arc length and angle in radians.
Central angle in radians
\(\theta = \dfrac{s}{r}\)Given arc length and radius.
Short Explanation
Use radians for arc length, and combine arc length with chord length for segment perimeter.
1. Arc Length Formula
When the central angle \(\theta\) is in radians, the arc length is \(s = r\theta\). If the angle is given in degrees, convert it to radians first by multiplying by \(\dfrac{\pi}{180^\circ}\), using \(\pi = 3.142\).
2. Rearranged Forms
Radius: \(r = \dfrac{s}{\theta}\). Central angle in radians: \(\theta = \dfrac{s}{r}\).
3. Reflex Angles
A reflex angle is between \(180^\circ\) and \(360^\circ\). For a major arc, convert the corresponding reflex angle to radians before using \(s = r\theta\).
4. Segment Perimeter
Perimeter of a segment \(= \text{arc length} + \text{chord length}\).
5. Chord Length — Cosine Rule
\(AC = \sqrt{2r^2 - 2r^2 \cos \theta}\). Ensure the calculator is in the correct angle mode when computing \(\cos \theta\).
6. Chord Length — Trigonometric Ratio
Drop a perpendicular from the centre to the chord. Then \(AC = 2r \sin\left(\dfrac{\theta}{2}\right)\).
| Quantity | Formula | Condition |
|---|---|---|
| Arc length | \(s = r\theta\) | \(\theta\) in radians |
| Radius | \(r = \dfrac{s}{\theta}\) | \(\theta\) in radians |
| Central angle | \(\theta = \dfrac{s}{r}\) | Answer in radians |
| Chord length | \(AC = 2r \sin\left(\dfrac{\theta}{2}\right)\) | Right-angled trigonometry |
| Chord length | \(AC = \sqrt{2r^2 - 2r^2 \cos \theta}\) | Cosine rule |
Steps for Finding the Perimeter of a Segment
Identify the radius \(r\) and central angle \(\theta\).
Convert \(\theta\) to radians if it is given in degrees.
Find the arc length using \(s = r\theta\).
Find the chord length, then add it to the arc length.
Try to Answer First
Answer in your mind, then press "Check Answer".
A sector of a circle has a radius of \(5\text{ cm}\) and subtends an angle of \(0.9\text{ rad}\) at the centre. What is the arc length of the sector?
Given an arc length \(s = 14\text{ cm}\) and radius \(r = 7\text{ cm}\), find the angle \(\theta\) subtended at the centre in radians, and convert it to degrees using \(\pi = 3.142\).
An arc subtends an angle of \(1.284\text{ rad}\) at the centre of a circle. If the major arc length is \(25\text{ cm}\), find the radius of the circle using \(\pi = 3.142\).
Common Mistakes
- Using the angle in degrees directly in \(s = r\theta\), for example \(5\text{ cm} \times 50^\circ\), without converting to radians first.
- Confusing segment perimeter with sector perimeter. Sector perimeter is \(2r + s\), while segment perimeter is arc length plus chord length.
- Forgetting to switch the calculator between Radian mode and Degree mode when computing \(\cos \theta\) or \(\sin\left(\dfrac{\theta}{2}\right)\).
- Using the minor central angle when the question asks for the major arc length, or vice versa.
Concept Misunderstandings
The formula \(s = r\theta\) works regardless of the unit of angle.
The formula \(s = r\theta\) is derived from the definition of a radian. If the angle is in degrees, use \(s = \dfrac{\theta}{360^\circ} \times 2\pi r\).
A chord and an arc with the same endpoints have equal length.
The chord is a straight line segment and is the shortest distance between two points on the circumference. The arc is curved, so arc length is always greater than chord length for any non-zero angle.
Summary
- The arc length formula in radians is \(s = r\theta\), where \(s\) is arc length, \(r\) is radius, and \(\theta\) is the central angle in radians.
- Radius is given by \(r = \dfrac{s}{\theta}\) and central angle by \(\theta = \dfrac{s}{r}\).
- Perimeter of a segment \(= \text{arc length} + \text{chord length}\).
- Chord length can be determined using the Cosine Rule, \(AC = \sqrt{2r^2 - 2r^2 \cos \theta}\), or trigonometric ratios, \(AC = 2r \sin\dfrac{\theta}{2}\).
- For real-world composite regions, identify all bounding curves and line segments carefully before summing up to find the total perimeter.
Short Activity
Objective: Check your understanding of arc length, segment perimeter, and circular measure formulae.
A. Objective Quiz
1 A circle has a radius of \(6\text{ cm}\) and an arc that subtends an angle of \(\dfrac{2}{3}\pi\text{ rad}\) at the centre. Using \(\pi = 3.142\), what is the length of the arc?
2 Which of the following expressions correctly represents the perimeter of a segment bounded by an arc of length \(s\) and a chord of length \(c\)?
B. Fill in the Blanks
3 In the circular measure formula \(s = r\theta\), the angle subtended at the centre \(\theta\) must be measured in ________.
4 In a circle of radius \(r\) where a chord subtends a central angle \(\theta\), the length of the chord can be calculated using the trigonometric formula \(2r\sin\left(\dfrac{________}{2}\right)\).
C. Matching / Drag and Drop
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
