We will learn the formula for finding the arc length of a sector. Consider “what fraction of the entire circle is represented by the central angle” to derive the formula naturally.
For a sector with radius r and central angle a degrees, the arc length ℓ is:
ℓ ＝ 2πr × (a/360)
Meaning: The arc of a sector is the part of the circumference (2πr) of a circle with radius r represented by the fraction a/360.
Example ①
Find the arc length of a sector with radius 6 cm and central angle 60°. Leave π as it is.
Solution:
ℓ = 2π × 6 × (60/360) = 12π × (1/6) = 2π (cm)
Find the arc length of a sector with radius 6 cm and central angle 60°. Leave π as it is.
Solution:
ℓ = 2π × 6 × (60/360) = 12π × (1/6) = 2π (cm)
Arc Lengths for Frequently Used Central Angles
It is quicker if you remember the “fraction of the circumference” for each central angle.
・30°：1/12 ・45°：1/8 ・60°：1/6 ・90°：1/4 ・120°：1/3
・180°：1/2 (semicircle) ・270°：3/4 ・360°：1 (full circle)
Example ②
For a sector with radius 8 cm, find the arc length for each central angle below. Leave π as it is.
(1) Central angle 90° (2) Central angle 135°
Solution:
Circumference = 2π×8 = 16π cm.
(1) ℓ = 16π × (90/360) = 16π × (1/4) = 4π (cm)
(2) ℓ = 16π × (135/360) = 16π × (3/8) = 6π (cm)
For a sector with radius 8 cm, find the arc length for each central angle below. Leave π as it is.
(1) Central angle 90° (2) Central angle 135°
Solution:
Circumference = 2π×8 = 16π cm.
(1) ℓ = 16π × (90/360) = 16π × (1/4) = 4π (cm)
(2) ℓ = 16π × (135/360) = 16π × (3/8) = 6π (cm)
Finding the Central Angle from the Arc Length
If the arc length is known, the central angle can be calculated in reverse.
a/360 ＝ ℓ / (2πr) → a ＝ 360 × ℓ/(2πr)
Example ③
A sector with radius 5 cm has an arc length of 2π cm. Find its central angle.
Solution:
Full circumference = 2π×5 = 10π cm.
a/360 = 2π/10π = 1/5 → a = 360 × 1/5 = 72°
A sector with radius 5 cm has an arc length of 2π cm. Find its central angle.
Solution:
Full circumference = 2π×5 = 10π cm.
a/360 = 2π/10π = 1/5 → a = 360 × 1/5 = 72°
Key Points
- Arc length ℓ = 2πr × (a/360)
- Remember it as “circumference × fraction represented by the central angle.”
- Remembering common fractions (90°→1/4, 60°→1/6) makes calculations faster.
- To find the central angle from an arc, use a = 360 × ℓ/(2πr).
Practice Problems
- Find the arc length of a sector with radius 9 cm and central angle 120°.
- Find the arc length of a sector with radius 6 cm and central angle 150°.
- A sector with radius 4 cm has an arc length of 3π cm. Find its central angle.
Answers and Explanations
- Answer: 6π cm
Explanation: ℓ = 2π×9×120/360 = 18π × 1/3 = 6π. - Answer: 5π cm
Explanation: ℓ = 2π×6×150/360 = 12π × 5/12 = 5π. - Answer: 135°
Explanation: Full circumference = 8π. a/360 = 3π/8π = 3/8 → a = 360 × 3/8 = 135°.