Arc Length of a Circle
The arc length of a circle is the distance along the curved line that forms part of the circumference. An arc is a portion of the circle's boundary, and its length depends on the radius of the circle and the central angle subtended by the arc.
This topic is part of the NCERT Class 10 chapter Areas Related to Circles. Arc length is closely related to the area of a sector — while the sector area measures the region enclosed, the arc length measures only the curved boundary.
Arc length problems frequently appear in board exams, often combined with sector area, segment area, and perimeter of sector calculations.
What is Arc Length of a Circle?
Definition: An arc is a continuous piece of a circle. The arc length is the distance measured along the arc from one endpoint to the other.
Types of Arcs:
- Minor Arc — the shorter arc subtending an angle theta < 180 degrees at the centre
- Major Arc — the longer arc subtending an angle (360 minus theta) at the centre
- Semicircular Arc — when theta = 180 degrees; arc length = half the circumference
Key Terms:
- theta — the central angle subtended by the arc (in degrees)
- r — radius of the circle
- Circumference — total arc length for the full circle = 2 pi r
- Chord — straight line joining the two endpoints of an arc (the chord is NOT the arc)
Arc Length of a Circle Formula
Arc Length Formula:
Arc Length = (theta / 360) x 2 pi r
Simplified:
Arc Length = (pi r theta) / 180
Major Arc Length:
Major Arc = 2 pi r minus Minor Arc Length
Perimeter of Sector:
Perimeter = Arc Length + 2r
Where:
- theta = central angle in degrees
- r = radius of the circle
- pi = 22/7 or 3.14159
Derivation and Proof
Derivation of Arc Length Formula:
- The circumference of a full circle = 2 pi r.
- A full circle subtends 360 degrees at the centre.
- An arc subtending theta degrees is a fraction theta / 360 of the full circle.
- Therefore, arc length = (theta / 360) x 2 pi r.
Verification with Standard Angles:
| Angle (theta) | Fraction of Circle | Arc Length |
|---|---|---|
| 360 deg | 360/360 = 1 | 2 pi r (full circumference) |
| 180 deg | 180/360 = 1/2 | pi r (semicircle) |
| 90 deg | 90/360 = 1/4 | pi r / 2 (quarter circle) |
| 60 deg | 60/360 = 1/6 | pi r / 3 |
Relationship between Arc Length and Sector Area:
- Arc Length = (theta/360) x 2 pi r
- Sector Area = (theta/360) x pi r squared
- Therefore: Sector Area = (1/2) x r x Arc Length
Types and Properties
Common Arc Length Problems:
| Problem Type | Given | Find |
|---|---|---|
| Find arc length | r and theta | (theta/360) x 2 pi r |
| Find major arc length | r and minor angle | 2 pi r minus minor arc |
| Find angle from arc length | r and arc length | theta = (arc x 360) / (2 pi r) |
| Find radius from arc length | theta and arc length | r = (arc x 360) / (2 pi theta) |
| Find perimeter of sector | r and theta | arc length + 2r |
| Clock problems | Hand length and time | Distance swept = arc length |
Solved Examples
Example 1: Basic Arc Length Calculation
Problem: Find the length of an arc subtending an angle of 90 degrees at the centre of a circle with radius 14 cm. (pi = 22/7)
Solution:
Given: r = 14 cm, theta = 90 deg
Formula: Arc length = (theta/360) x 2 pi r
- = (90/360) x 2 x (22/7) x 14
- = (1/4) x 2 x 22 x 2
- = (1/4) x 88
- = 22 cm
Answer: Arc length = 22 cm
Example 2: Arc Length with Angle 60 Degrees
Problem: Find the arc length when r = 21 cm and theta = 60 degrees. (pi = 22/7)
Solution:
Given: r = 21 cm, theta = 60 deg
- Arc = (60/360) x 2 x (22/7) x 21
- = (1/6) x 2 x 22 x 3
- = (1/6) x 132
- = 22 cm
Answer: Arc length = 22 cm
Example 3: Major and Minor Arc Lengths
Problem: A chord subtends an angle of 120 degrees at the centre of a circle with radius 7 cm. Find the lengths of the minor and major arcs. (pi = 22/7)
Solution:
Given: r = 7 cm, theta (minor) = 120 deg
Minor arc:
- = (120/360) x 2 x (22/7) x 7
- = (1/3) x 44
- = 14.67 cm
Major arc:
- Major angle = 360 minus 120 = 240 deg
- = (240/360) x 44 = (2/3) x 44 = 29.33 cm
Verification: 14.67 + 29.33 = 44 = 2 pi r ✓
Answer: Minor arc = 14.67 cm, Major arc = 29.33 cm
Example 4: Finding the Angle from Arc Length
Problem: The arc length of a sector is 11 cm and the radius is 7 cm. Find the angle of the sector. (pi = 22/7)
Solution:
Given: Arc = 11 cm, r = 7 cm
Formula: Arc = (theta/360) x 2 pi r
- 11 = (theta/360) x 2 x (22/7) x 7
- 11 = (theta/360) x 44
- theta/360 = 11/44 = 1/4
- theta = 360/4 = 90 degrees
Answer: Angle = 90 degrees
Example 5: Finding Radius from Arc Length
Problem: An arc subtending 72 degrees at the centre has length 44 cm. Find the radius. (pi = 22/7)
Solution:
Given: Arc = 44 cm, theta = 72 deg
Formula: 44 = (72/360) x 2 x (22/7) x r
- 44 = (1/5) x (44/7) x r
- 44 = (44r)/35
- r = (44 x 35)/44 = 35 cm
Answer: Radius = 35 cm
Example 6: Perimeter of a Sector
Problem: Find the perimeter of a sector with radius 10.5 cm and angle 60 degrees. (pi = 22/7)
Solution:
Given: r = 10.5 cm, theta = 60 deg
Step 1: Arc length = (60/360) x 2 x (22/7) x 10.5 = (1/6) x 66 = 11 cm
Step 2: Perimeter = arc + 2r = 11 + 21 = 32 cm
Answer: Perimeter of sector = 32 cm
Example 7: Clock Problem — Distance Covered by Tip
Problem: The minute hand of a clock is 14 cm long. Find the distance covered by its tip in 30 minutes. (pi = 22/7)
Solution:
Given: r = 14 cm (length of minute hand)
Step 1: In 30 minutes, the minute hand rotates 180 degrees (half the clock face).
Step 2: Arc length = (180/360) x 2 x (22/7) x 14 = (1/2) x 88 = 44 cm
Answer: Distance covered = 44 cm
Example 8: Clock Problem — 15 Minutes
Problem: The minute hand of a clock is 7 cm long. Find the distance the tip moves in 15 minutes. (pi = 22/7)
Solution:
Given: r = 7 cm
Step 1: In 15 minutes, angle swept = (15/60) x 360 = 90 degrees
Step 2: Arc = (90/360) x 2 x (22/7) x 7 = (1/4) x 44 = 11 cm
Answer: Distance = 11 cm
Example 9: Wire Bent into an Arc
Problem: A wire of length 33 cm is bent into an arc of a circle of radius 21 cm. Find the angle subtended by the arc at the centre. (pi = 22/7)
Solution:
Given: Arc = 33 cm, r = 21 cm
Formula: 33 = (theta/360) x 2 x (22/7) x 21
- 33 = (theta/360) x 132
- theta/360 = 33/132 = 1/4
- theta = 90 degrees
Answer: Angle = 90 degrees
Example 10: Sector Area from Arc Length
Problem: The arc length of a sector is 15 cm and the radius is 10 cm. Find the area of the sector.
Solution:
Given: Arc = 15 cm, r = 10 cm
Using the relationship: Sector Area = (1/2) x r x Arc Length
- Area = (1/2) x 10 x 15 = 75 sq cm
Verification: theta = (Arc x 360)/(2 pi r) = (15 x 360)/(20 pi) = 5400/(20 x 3.14159) = 85.94 deg
Area = (85.94/360) x pi x 100 = 0.2387 x 314.159 = 75 sq cm ✓
Answer: Sector area = 75 cm squared
Real-World Applications
Real-life applications of arc length:
- Wheel Distance: The distance a wheel covers in one rotation equals its circumference (arc for full 360 degrees). Partial rotations use the arc length formula.
- Clock Hands: The distance covered by the tip of a clock hand in a given time period is an arc length problem.
- Roller Coasters: The track length of curved sections is calculated using arc lengths.
- Roads and Railways: Curved sections of roads and railway tracks are designed using arc length calculations.
- Protractors and Instruments: Markings on circular instruments use arc length for equal spacing.
- Satellite Orbits: The distance a satellite covers along its circular orbit is an arc length calculation.
Key Points to Remember
- Arc length = (theta/360) x 2 pi r.
- Simplified: Arc length = (pi r theta) / 180.
- Major arc = circumference minus minor arc = 2 pi r minus minor arc.
- Perimeter of sector = arc length + 2r (arc plus two radii).
- Sector Area = (1/2) x r x arc length.
- To find angle from arc: theta = (arc x 360) / (2 pi r).
- To find radius from arc: r = (arc x 180) / (pi theta).
- Minor arc + Major arc = circumference = 2 pi r.
- The minute hand sweeps 6 degrees per minute (360/60). The hour hand sweeps 0.5 degrees per minute (360/720).
- Arc length is measured in linear units (cm, m), NOT in square units.
Practice Problems
- Find the arc length for a circle of radius 28 cm with central angle 45 degrees. (pi = 22/7)
- The arc length is 33 cm and the radius is 42 cm. Find the central angle.
- Find the minor and major arc lengths when r = 14 cm and the central angle is 150 degrees.
- The minute hand of a clock is 21 cm long. Find the distance covered by its tip in 40 minutes.
- A sector has perimeter 31 cm and arc length 11 cm. Find the radius.
- A wire 66 cm long is bent into a circle. Find the radius. If the same wire is bent into a sector of angle 60 degrees, find the radius of the sector.
- Find the area of a sector whose arc length is 22 cm and radius is 14 cm.
Frequently Asked Questions
Q1. What is arc length?
Arc length is the distance measured along the curved part of a circle between two points. It is a portion of the circumference. Arc length = (theta/360) x 2 pi r.
Q2. What is the difference between arc length and chord length?
Arc length is measured along the curve. Chord length is the straight-line distance between the two endpoints of the arc. Arc length is always greater than or equal to chord length.
Q3. What is the formula for the perimeter of a sector?
Perimeter of a sector = arc length + 2r. It includes the curved part (arc) and two straight edges (radii).
Q4. How is arc length related to sector area?
Sector Area = (1/2) x r x arc length. This is analogous to the triangle area formula (1/2 x base x height), where the arc acts like the base and the radius acts like the height.
Q5. How do you find the arc length for a clock hand?
The minute hand moves 360 degrees in 60 minutes (6 degrees per minute). Multiply the number of minutes by 6 to get theta, then use arc = (theta/360) x 2 pi r, where r = length of the hand.
Q6. Can arc length be greater than the circumference?
No. The maximum arc length equals the full circumference (when theta = 360 degrees). Since 0 ≤ theta ≤ 360, arc length ranges from 0 to 2 pi r.
Q7. How do you find the angle when arc length and radius are given?
Use theta = (arc length x 360) / (2 pi r). Substitute the known values of arc and r to find the central angle in degrees.
Q8. What units is arc length measured in?
Arc length is a length (not area), so it is measured in linear units like cm, m, km. Do NOT write sq cm for arc length.
Q9. Is the minor arc always shorter than the major arc?
Yes. The minor arc corresponds to the smaller central angle (< 180 deg), and the major arc to the larger angle (> 180 deg). When theta = 180, both arcs are equal (each is a semicircle).
Q10. What is the arc length formula in radians?
In radians, arc length = r x theta (where theta is in radians). This is simpler but is not used in CBSE Class 10 — the degree-based formula is used instead.
