Arc length is the distance along a curved line between two points on a circle or other curve
The arc length formula depends on whether you are working with a circle or a more complex curve. For a circle, you need the radius and the central angle (the angle at the center of the circle that the arc spans). For other curves, you need calculus. This guide covers the circle formula first, then shows how the calculus version works so you understand what it measures.
The most common case — a circular arc — uses this formula: arc length = radius × angle in radians. If your angle is in degrees, you convert it to radians first by multiplying by π/180. Once you have the radius and angle in radians, multiply them together and you have the arc length.
Key Takeaways
- For a circular arc, multiply the radius by the central angle measured in radians: arc length = r × θ.
- If your angle is in degrees, convert it to radians by multiplying by π/180 before using the formula.
- The arc length formula for curves beyond circles requires calculus and integration, but the principle is the same: you are measuring distance along a path.
- A radian is a unit of angle where one radian equals the angle that produces an arc length equal to the radius itself.
The circular arc length formula and how to use it
Start with the formula: s = rθ, where s is arc length, r is the radius, and θ (theta) is the central angle in radians. This formula works because a radian is defined so that an arc of one radian always has length equal to the radius. So if you have a 2-radian angle on a circle with radius 5, the arc length is 5 × 2 = 10 units.
To use this formula, you need two pieces of information: the radius of the circle and the central angle. The radius is the distance from the center of the circle to any point on the circle. The central angle is the angle formed by two radii that touch the endpoints of your arc. Measure this angle at the center of the circle, not at the edge.
If you are given the angle in degrees instead of radians, convert first. Multiply the degree measure by π/180. For example, a 90-degree angle becomes 90 × (π/180) = π/2 radians, which equals about 1.57 radians. Then use the formula with the radian measure.
Converting degrees to radians
Radians and degrees are two ways to measure the same angle. A full circle is 360 degrees or 2π radians. To convert any angle from degrees to radians, use this formula: radians = degrees × (π/180).
Here are some common conversions you may see:
| Degrees | Radians | Decimal (approx.) |
|---|---|---|
| 30° | π/6 | 0.52 |
| 45° | π/4 | 0.79 |
| 60° | π/3 | 1.05 |
| 90° | π/2 | 1.57 |
| 180° | π | 3.14 |
| 360° | 2π | 6.28 |
If you have a calculator, you can enter the degree value, multiply by π/180, and get a decimal. If you are working by hand and the angle is a common one like 45 or 60 degrees, it is often easier to leave the answer in terms of π (like π/4 radians) until the final step.
Worked example: finding arc length on a circle
Suppose you have a circle with radius 8 cm and you want to find the arc length of an arc that spans a central angle of 60 degrees. First, convert 60 degrees to radians: 60 × (π/180) = π/3 radians. Then use the formula: arc length = 8 × (π/3) = 8π/3 cm. If you want a decimal, 8π/3 ≈ 8.38 cm.
Here is another example with a larger angle. A circle has radius 10 meters and the arc spans 120 degrees. Convert: 120 × (π/180) = 2π/3 radians. Arc length = 10 × (2π/3) = 20π/3 meters ≈ 20.94 meters. Notice that doubling the angle (from 60 to 120 degrees) doubles the arc length (from 8π/3 to 20π/3). This relationship holds because the formula is linear in the angle.
Arc length for curves beyond circles
When the curve is not a circle — for example, a parabola, an ellipse, or a spiral — you cannot use the straightforward formula s = rθ. Instead, you use calculus. The arc length formula for a curve defined by a function y = f(x) from x = a to x = b is the integral: s = ∫[a to b] √(1 + (dy/dx)²) dx.
This formula works by breaking the curve into tiny straight segments, finding the length of each segment using the Pythagorean theorem, and adding them all up. The derivative dy/dx tells you the slope of the curve at each point. The square root expression √(1 + (dy/dx)²) is the length of a tiny segment of the curve. The integral symbol ∫ means you are adding up infinitely many of these tiny segments.
Computing this integral by hand requires knowing calculus techniques like substitution or integration by parts. For most non-circular curves, you will use a computer or graphing calculator to evaluate the integral. The important thing to understand is that the method is the same: measure the distance along the path by summing infinitesimal pieces.
When you need arc length in real situations
Arc length comes up in engineering, physics, and design. If you are laying out a curved road or railroad track, you need to know the actual distance along the curve, not the straight-line distance between the endpoints. If you are wrapping a cable around a drum or spool, the arc length tells you how much cable fits. In physics, if an object moves along a curved path, the arc length is the distance it actually travels.
In most practical cases, you will either be given the radius and angle (and use s = rθ), or you will have a curve defined by an equation (and use calculus or a calculator). Understanding which formula applies to your situation is the first step.
Frequently Asked Questions
What is the difference between arc length and the chord?
The arc is the curved path along the circle between two points. The chord is the straight line connecting those same two points. The arc length is always longer than the chord length, unless the arc is very small. For a semicircle (180 degrees), the arc length is πr and the chord length is 2r, so the arc is longer by a factor of π/2 ≈ 1.57.
Can I use the arc length formula if I know the chord length instead of the angle?
Not directly. If you know the chord length and the radius, you can find the central angle using trigonometry, then use the arc length formula. The relationship is: chord = 2r sin(θ/2), where θ is the central angle in radians. Rearrange to solve for θ, then use s = rθ.
Why do I have to convert degrees to radians?
The formula s = rθ only works when θ is in radians. This is because a radian is defined so that the arc length equals the radius when the angle is 1 radian. Degrees do not have this property, so the formula would give wrong answers if you used degrees directly.
What if the arc wraps more than once around the circle?
The formula still works. If the central angle is 720 degrees (two full rotations), convert to radians: 720 × (π/180) = 4π radians. Then multiply by the radius. An arc that goes around twice is straightforward twice as long as an arc that goes around once.
Do I need calculus to find arc length for any curve?
For circles and circular arcs, no — the straightforward formula s = rθ is all you need. For any other smooth curve, yes, you need the calculus integral. Some special curves (like ellipses) have arc length formulas, but they are derived using calculus and are more complex than the circular case.