How to Find Arc Length Using Radius and Angle?

Arc Length is the distance measured along a curved part of a circle. It depends on the circle’s radius and the central angle. This page covers arc length formulas in degrees and radians, sector perimeter, major and minor arcs, unit conversions, solved examples and quick methods for aptitude problems.

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What Is Arc Length?

Arc length is the distance between two points measured along the circumference of a circle. For radius r and central angle θ, it is calculated using the fraction of the full circumference intercepted by the angle.

For an angle θ measured in degrees, arc length is L = (θ/360°) × 2πr. The full circumference corresponds to 360°, so a semicircle has length πr and a quarter-circle has length πr/2. When θ is measured in radians, the formula is L = rθ.

Arc Length Video Lessons

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Circle Arc Length: Formula and Examples

Learn how to calculate the arc length of a circle using the central angle and radius, with clear formulas and worked examples for quantitative aptitude.

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Arc Length Formula & Tricks

Important Formulas

Arc length in degrees
L = (θ/360°) × 2πr

Use this formula when the central angle θ is given in degrees, r is the radius, and L is the arc length.

Arc length in radians
L = rθ

Use this formula when θ is measured in radians.

Degree-radian conversion
θ radians = θ° × π/180

Convert a degree measure to radians before applying L = rθ.

Sector perimeter
P = 2r + L

The perimeter of a sector consists of its two radii and the arc joining their endpoints.

Quick Tricks

Use the angle fraction of the circumference

Find θ/360 first, then multiply it by 2πr. This avoids expanding the full formula unnecessarily.

Example: For r = 14 cm and θ = 90°, L = (90/360) × 2π × 14 = 7π = 22 cm when π = 22/7.
Simplify before substituting π

Cancel the angle fraction with 2r before using π. This usually makes calculations faster.

Example: For r = 21 cm and θ = 120°, L = (120/360) × 2π × 21 = 14π = 44 cm when π = 22/7.

Arc Length Concepts

Arc Length for Angles in Degrees

For a central angle measured in degrees, arc length is L = (θ/360°) × 2πr.

The angle forms θ/360 of one complete revolution. Therefore, the corresponding arc is the same fraction of the circumference 2πr. For r = 7 cm and θ = 180°, L = (180/360) × 2π × 7 = 7π cm, or 22 cm when π = 22/7.

Example: If r = 10 cm and θ = 72°, L = (72/360) × 2π × 10 = 4π cm.

Arc Length for Angles in Radians

When the central angle is in radians, arc length is L = rθ.

One complete revolution is 2π radians. Since the circumference is 2πr, an angle of θ radians intercepts a length equal to rθ. For example, r = 8 cm and θ = π/3 give L = 8π/3 cm.

Example: If r = 6 m and θ = 2π/3, then L = 6 × 2π/3 = 4π m.

Converting Degrees to Radians

To convert θ degrees to radians, multiply by π/180.

Common conversions are 90° = π/2, 180° = π, 270° = 3π/2 and 360° = 2π. After conversion, use L = rθ. For 60°, θ = 60π/180 = π/3.

Example: For r = 12 cm and an angle of 60°, L = 12 × π/3 = 4π cm.

Perimeter of a Sector

The perimeter of a sector is the arc length plus the lengths of its two radii: P = L + 2r.

The two radii are straight sides and the arc is the curved side. For r = 7 cm and θ = 90°, the arc length is 7π cm. Hence, sector perimeter = 7π + 14 cm, which is 36 cm when π = 22/7.

Example: For r = 14 cm and θ = 60°, L = 14π/3 cm, so the sector perimeter is 14 + 14π/3 cm.

Minor Arc, Major Arc and Full Circumference

A minor arc corresponds to an angle less than 180°, a semicircular arc corresponds to 180°, and a major arc generally corresponds to an angle greater than 180°.

For a major arc, use its major central angle directly or subtract the minor arc length from the full circumference. If a minor arc has angle 80°, the major arc has angle 280°. Their lengths together equal 2πr.

Example: For r = 9 cm and a minor angle of 80°, the major arc length is (280/360) × 2π × 9 = 14π cm.

Arc Length Visual Guide & Data

Use the table and chart below to understand the data better.

Common Central Angles and Arc Lengths

Arc lengths shown as fractions or multiples of the radius r for a circle of radius r.

Central angle Arc length
90° πr/2
120° 2πr/3
180° πr
270° 3πr/2
360° 2πr

Note: These results follow from L = (θ/360°) × 2πr.

Central Angle and Corresponding Arc

Visual guide: Central Angle and Corresponding Arc
Radius: r Central angle: θ Arc length: L
A sector shows radius r, central angle θ and the curved arc whose length is L.

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Practice Arc Length Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Arc Length Revision Points

Use the angle and radius to select the correct arc length formula.

  • In degrees, L = (θ/360°) × 2πr.
  • In radians, L = rθ.
  • Convert degrees to radians using θ × π/180.
  • A full circle has arc length 2πr.
  • A semicircle has arc length πr.
  • Sector perimeter = arc length + 2r.
  • Major arc length and minor arc length add up to the full circumference.
  • Use consistent units for radius and arc length.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Arc Length FAQs

What is the arc length formula when the angle is in degrees?

The formula is L = (θ/360°) × 2πr, where θ is the central angle in degrees and r is the radius.

What is the arc length formula when the angle is in radians?

The formula is L = rθ, where r is the radius and θ is the angle in radians.

Find the arc length for r = 14 cm and θ = 90°.

L = (90/360) × 2π × 14 = 7π cm = 22 cm, using π = 22/7.

Find the arc length for r = 10 cm and θ = π/5 radians.

Using L = rθ, L = 10 × π/5 = 2π cm.

How is the perimeter of a sector calculated?

Sector perimeter = arc length + 2r. The two radii are added to the curved arc.

What is the arc length of a semicircle of radius 7 cm?

A semicircle has angle 180°, so L = πr = 7π cm = 22 cm when π = 22/7.

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