Length of an ARC : āĻŦā§āĻ¤ā§āĻ¤ā§° āĻĒā§°āĻŋāĻ§āĻŋā§° āĻāĻāĻž āĻ āĻāĻļā§° āĻĻā§ā§°ā§āĻā§āĻ¯
1. What is an Arc ?
An arc is a portion of the circumference of a circle. : Arc (āĻāĻžāĻĒ) āĻšā§āĻā§ āĻŦā§āĻ¤ā§āĻ¤ā§° āĻĒā§°āĻŋāĻ§āĻŋā§° āĻāĻāĻž āĻ
āĻāĻļāĨ¤
2. Formula for Length of an ARC
Arc Length (āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯) = θ / 360â × 2πr
Where:
- θ (theta) = central angle (in degrees) : āĻā§āĻ¨ā§āĻĻā§ā§°ā§āĻ¯āĻŧ āĻā§āĻŖ (āĻĄāĻŋāĻā§ā§°ā§)
- r = radius of the circle : āĻ ā§°ā§āĻ§āĻŦā§āĻ¯āĻžāĻ¸
- Unit (āĻāĻāĻ) = cm, m, etc.
Important Points (āĻā§ā§°ā§āĻ¤ā§āĻŦāĻĒā§āĻ°ā§āĻŖ āĻāĻĨāĻž)
- Full circle (360°) → Arc length = 2πr (circumference) : āĻ¸āĻŽā§āĻĒā§ā§°ā§āĻŖ āĻŦā§āĻ¤ā§āĻ¤ (360°) → āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ = 2πr
- Arc is always a part of the circle : Arc āĻ¸āĻĻāĻžāĻ¯āĻŧ āĻŦā§āĻ¤ā§āĻ¤ā§° āĻāĻāĻž āĻ āĻāĻļ
- Larger angle → larger arc : āĻĄāĻžāĻā§° āĻā§āĻŖ → āĻĄāĻžāĻā§° āĻāĻžāĻĒ
Quick Revision (āĻĻā§ā§°ā§āĻ¤ āĻĒā§āĻ¨ā§°āĻžāĻŦā§āĻ¤ā§āĻ¤āĻŋ)
- Arc = part of circle (āĻŦā§āĻ¤ā§āĻ¤ā§° āĻ āĻāĻļ)
- Formula → (θ/360°) × 2πr
- 360° → full circumference
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PYQ MCQ
Q1. Find the length of arc of a circle of radius 7 cm and central angle 60°. : āĻĒā§ā§°āĻļā§āĻ¨: 7 cm āĻ ā§°ā§āĻ§āĻŦā§āĻ¯āĻžāĻ¸ āĻā§°ā§ 60° āĻā§āĻ¨ā§āĻĻā§ā§°ā§āĻ¯āĻŧ āĻā§āĻŖā§° āĻŦā§āĻ¤ā§āĻ¤ā§° āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 22/3 cm (b) 14/3 cm (c) 44/3 cm (d) 11 cm
Ans: (a) 22/3 cm
Explanation : Arc Length = (θ/360) × 2πr = (60/360) × 2 × 22/7 × 7 = 22/3 cm
Q2. Find arc length when radius = 14 cm and angle = 90°. āĻĒā§ā§°āĻļā§āĻ¨: r = 14 cm āĻā§°ā§ āĻā§āĻŖ = 90° āĻšāĻ˛ā§ āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 22 cm (b) 44 cm (c) 66 cm (d) 88 cm
Ans: (a) 22 cm
Explanation: = (90/360) × 2 × 22/7 × 14 = 22 cm
Q3. If the central angle is 180°, what fraction of circumference is the arc ? : āĻĒā§ā§°āĻļā§āĻ¨: āĻā§āĻ¨ā§āĻĻā§ā§°ā§āĻ¯āĻŧ āĻā§āĻŖ 180° āĻšāĻ˛ā§ āĻāĻžāĻĒāĻā§ āĻĒā§°āĻŋāĻ§āĻŋā§° āĻāĻŋāĻŽāĻžāĻ¨ āĻ āĻāĻļ ?
Options: (a) 1/4 (b) 1/2 (c) 3/4 (d) Full
Ans: (b) 1/2
Explanation: 180° / 360° = 1/2
Q4. Find arc length if circumference is 44 cm and angle is 90°. : āĻĒā§ā§°āĻļā§āĻ¨: āĻĒā§°āĻŋāĻ§āĻŋ 44 cm āĻā§°ā§ āĻā§āĻŖ 90° āĻšāĻ˛ā§ āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 11 cm (b) 22 cm (c) 33 cm (d) 44 cm
Ans: (a) 11 cm
Explanation: Arc = (90/360) × 44 = 11 cm
Q5. Find arc length of full circle. āĻĒā§ā§°āĻļā§āĻ¨: āĻ¸āĻŽā§āĻĒā§ā§°ā§āĻŖ āĻŦā§āĻ¤ā§āĻ¤ā§° āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) πr² (b) 2πr (c) πr (d) r²
Ans: (b) 2πr
Explanation: Full circle (360°) → Arc = Circumference = 2πr : āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: 360° āĻšāĻ˛ā§ → āĻāĻžāĻĒ = āĻĒā§°āĻŋāĻ§āĻŋ = 2πr
Q6. Radius = 21 cm, angle = 120°. Find arc length. : āĻĒā§ā§°āĻļā§āĻ¨: r = 21 cm āĻā§°ā§ āĻā§āĻŖ = 120° āĻšāĻ˛ā§ āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 44 cm (b) 88 cm (c) 66 cm (d) 132 cm
Ans: (b) 88 cm
Explanation: = (120/360) × 2 × 22/7 × 21 = 88 cm
Q7. If angle doubles, arc length becomes ? āĻĒā§ā§°āĻļā§āĻ¨: āĻā§āĻŖ āĻĻā§āĻā§āĻŖ āĻā§°āĻŋāĻ˛ā§ āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋ āĻšāĻ¯āĻŧ ?
Options: (a) Same (b) Double (c) Half (d) Four times
Ans: (b) Double
Explanation: Arc length ∝ angle āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ ∝ āĻā§āĻŖ
Q8. Find arc length when θ = 30°, r = 14 cm. : āĻĒā§ā§°āĻļā§āĻ¨: θ = 30°, r = 14 cm āĻšāĻ˛ā§ āĻāĻžāĻĒā§° āĻĻā§ā§°ā§āĻā§āĻ¯ āĻāĻŋāĻŽāĻžāĻ¨ ?
Options: (a) 22/3 cm (b) 44/3 cm (c) 11/3 cm (d) 7 cm
Ans: (c) 11/3 cm
Explanation: = (30/360) × 2 × 22/7 × 14 = 11/3 cm āĻŦā§āĻ¯āĻžāĻā§āĻ¯āĻž: = (30/360) × 2 × 22/7 × 14 = 11/3 cm
Exam Tricks (āĻā§ā§°ā§āĻ¤ā§āĻŦāĻĒā§āĻ°ā§āĻŖ āĻāĻŋāĻĒāĻ)
- 60° → 1/6 of circle
- 90° → 1/4
- 180° → 1/2
- 360° → Full