Arc Length Calculator
An arc length calculator finds the distance along a curved part of a circle when the radius and central angle are known. Arc length is commonly used in geometry, engineering, construction, design, and calculations involving circular paths.
What Is Arc Length?
An arc is a portion of the circumference of a circle. The arc length is the distance measured along that curved portion rather than the straight-line distance between its endpoints.
Arc Length Formula
When the central angle is measured in degrees, the formula is:
L = (θ ÷ 360) × 2πr
Here, L is the arc length, θ is the central angle in degrees, and r is the radius of the circle.
How to Calculate Arc Length
Suppose a circle has a radius of 10 cm and a central angle of 90°.
Substitute the values:
L = (90 ÷ 360) × 2π × 10
L = 5π ≈ 15.71 cm
Therefore, the arc length is approximately 15.71 centimeters.
Arc Length Using Radians
If the central angle is given in radians, the formula is simpler:
L = rθ
For example, if the radius is 6 meters and the angle is π/3 radians:
L = 6 × π/3 = 2π ≈ 6.28 m
Common Arc Lengths
| Central Angle | Fraction of Circle | Arc Length |
|---|---|---|
| 30° | 1/12 | πr ÷ 6 |
| 60° | 1/6 | πr ÷ 3 |
| 90° | 1/4 | πr ÷ 2 |
| 180° | 1/2 | πr |
| 360° | Full circle | 2πr |
Arc Length vs. Chord Length
Arc length and chord length are different measurements. Arc length follows the curved edge of the circle, while chord length is the straight-line distance between the two endpoints of the arc.
For the same two points on a circle, the arc length is generally longer than the corresponding chord.
Minor and Major Arcs
A minor arc measures less than 180°, while a major arc measures more than 180°. A semicircle has an arc measure of exactly 180°.
The same arc length formula can be used for either type as long as the correct central angle is entered.
Where Is Arc Length Used?
Arc length calculations are useful for circular tracks, curved structures, mechanical parts, wheels, pipes, architectural designs, and geometry problems involving portions of a circle.
Common Mistakes to Avoid
Make sure you use the radius rather than the diameter in the formula. If the diameter is given, divide it by 2 first.
Also check whether the angle is in degrees or radians. The degree formula and radian formula use different forms.
Quick Summary
Arc length measures the distance along a curved portion of a circle. For an angle in degrees, use L = (θ ÷ 360) × 2πr. For an angle in radians, use L = rθ. The final answer is expressed in the same linear units as the radius.