Arc Length of a Circle: Formula, Examples, and Step-by-Step Guide

Arc Length of a Circle: Formula, Examples, and Step-by-Step Guide

A curved distance is still a distance, and you can measure it with two numbers.

Think of the crust on a slice of pizza, or the bend of a running track. Both are curves with a length you can measure if you know the size of the circle and how much of it the curve covers. This guide shows you how to calculate that length from the radius and the central angle, in degrees and in radians, with worked examples. Check any answer with our Arc Length Calculator, and see how to find arc length without calculus for curves that are not circles.

What Is the Arc Length of a Circle?

Arc length is the distance you travel along the curve itself.

An arc is a piece of a circle between two points, and its arc length is the distance along the curve from one point to the other. The radius is the distance from the centre to any point on the circle. The central angle is the angle at the centre between the two radii that reach the ends of the arc. A bigger radius or a wider angle both give a longer arc.

Arc length is not chord length. The chord is the straight line joining the two end points, and it cuts across the bend, so it is always shorter than the arc. Arc length is a part of the circumference, the distance all the way round the circle: an arc covering half the circle is half the circumference, and one covering a quarter is a quarter of it.

A pizza slice with radius $15$ cm and a central angle of $60^\circ$ covers one sixth of the pizza, so its crust is $L = \frac{60}{360} \times 2\pi \times 15 = 5\pi \approx 15.71$ cm.

What Is the Formula for Arc Length?

Two formulas cover every circle problem: one for degrees, one for radians.

Degrees:

$L = \frac{\theta}{360^\circ} \times 2\pi r$

Radians:

$L = r\theta$

Here $L$ is the arc length, $r$ is the radius, $\theta$ is the central angle, and $\pi$ is about $3.14159$. The degree formula takes a fraction of the full circumference $2\pi r$, and the radian formula is shorter because a radian already measures the angle in radius lengths.

Method	Formula	Use it when
Degrees	$L = \frac{\theta}{360^\circ} \times 2\pi r$	The angle is given in degrees
Radians	$L = r\theta$	The angle is given in radians

The degree formula divides by $360$ because a full circle turns through $360$ degrees. The radian formula has no $360$ because a full circle is $2\pi$ radians. Our Circle Arc Length Calculator handles both.

How to Find Arc Length Step by Step

Five steps solve every basic problem.

1. Identify the radius. If you are given the diameter, halve it first.
2. Identify the central angle.
3. Check whether the angle is in degrees or radians.
4. Choose the correct formula for that unit.
5. Substitute the values and calculate.

Short demo: radius $10$ cm, angle $72^\circ$. In degrees, $L = \frac{72}{360} \times 2\pi \times 10 = \frac{1}{5} \times 20\pi = 4\pi \approx 12.57$ cm.

How to Calculate Arc Length Using Degrees

Degrees are the unit most school problems use.

Work through the fraction first, then multiply by the circumference, keeping $\pi$ in the answer until the last step.

Example 1: Radius $7$ cm, central angle $90^\circ$.

1. Substitute: $L = \frac{90}{360} \times 2\pi \times 7$.
2. Simplify the fraction: $\frac{90}{360} = \frac{1}{4}$, because $90^\circ$ is one quarter of a full circle.
3. Circumference: $2\pi \times 7 = 14\pi$.
4. Multiply: $L = \frac{1}{4} \times 14\pi = \frac{7\pi}{2} \approx 11.00$ cm.

Example 2: Radius $12$ metres, central angle $150^\circ$.

1. Substitute: $L = \frac{150}{360} \times 2\pi \times 12$.
2. Simplify the fraction: $\frac{150}{360} = \frac{5}{12}$.
3. Circumference: $2\pi \times 12 = 24\pi$.
4. Multiply: $L = \frac{5}{12} \times 24\pi = 10\pi \approx 31.42$ metres.

How to Calculate Arc Length Using Radians

Radians shorten the formula to a single multiplication.

A radian is the angle you get when the arc length equals the radius. Since the circumference is $2\pi r$, a full circle is $2\pi$ radians and a half circle is $\pi$ radians. To change degrees into radians, multiply by $\frac{\pi}{180}$:

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Degrees	Radians
$30^\circ$	$\frac{\pi}{6}$
$45^\circ$	$\frac{\pi}{4}$
$60^\circ$	$\frac{\pi}{3}$
$90^\circ$	$\frac{\pi}{2}$
$180^\circ$	$\pi$
$360^\circ$	$2\pi$

Worked example: Radius $6$ cm, angle $\frac{2\pi}{3}$ radians. Use $L = r\theta = 6 \times \frac{2\pi}{3} = 4\pi \approx 12.57$ cm.

The radian formula is simpler when the angle is already in radians, and it is the formula used in trigonometry and physics. Compare the two methods below.

How to Find Arc Length When the Diameter Is Given

Halve the diameter first, then use the usual formula.

The formulas use the radius, not the diameter. The radius is half the diameter:

$r = \frac{d}{2}$

Worked example: Diameter $14$ cm, central angle $60^\circ$.

1. Radius: $r = \frac{14}{2} = 7$ cm.
2. Substitute into the degree formula: $L = \frac{60}{360} \times 2\pi \times 7$.
3. Simplify: $\frac{60}{360} = \frac{1}{6}$, and $2\pi \times 7 = 14\pi$.
4. Multiply: $L = \frac{1}{6} \times 14\pi = \frac{7\pi}{3} \approx 7.33$ cm.

How to Find Arc Length for Major and Minor Arcs

One circle, two arcs: the short way round and the long way round.

Two points on a circle split it into two arcs. The minor arc is the shorter one, with a central angle under $180^\circ$. The major arc is the longer one, with a central angle over $180^\circ$. A semicircle is exactly half the circle, with an arc length of $\pi r$. The two arcs add up to the full circumference, so the major arc is the circumference minus the minor arc:

$L_{\text{major}} = 2\pi r - L_{\text{minor}}$

Try the selector below, then work through the example.

Worked example: Radius $10$ cm, minor arc angle $120^\circ$.

1. Minor arc: $L = \frac{120}{360} \times 2\pi \times 10 = \frac{20\pi}{3} \approx 20.94$ cm.
2. Circumference: $C = 2\pi \times 10 = 20\pi \approx 62.83$ cm.
3. Major arc: $L = 62.83 - 20.94 = 41.89$ cm. Check: the major angle is $360^\circ - 120^\circ = 240^\circ$, and $\frac{240}{360} \times 20\pi = \frac{40\pi}{3} \approx 41.89$ cm. Both routes agree.

A semicircle is the simplest case of all; see our Semicircle Arc Length Calculator.

Common Mistakes When Finding Arc Length

Seven small slips cause most wrong answers.

• Using diameter instead of radius. The formulas use $r$. If a problem gives a diameter of $20$, halve it to $r = 10$ first, or your answer doubles.
• Mixing degrees and radians. An angle of $90^\circ$ is not $90$ radians. Check the unit before you pick a formula, and check your calculator mode too.
• Forgetting to divide the angle by 360. In the degree formula, $\frac{\theta}{360}$ is the fraction of the circle. Skipping it gives the whole circumference instead of the arc.
• Confusing arc length with chord length. The chord cuts straight across, so it is always shorter than the arc between the same two points.
• Using the wrong central angle. A problem may ask for the major arc while giving the minor angle. Subtract from $360^\circ$ (or $2\pi$ radians) when you need the other arc.
• Forgetting units. Arc length is a length, so it needs cm, m, or inches, just like the radius.
• Rounding too early. Keep $\pi$ in the answer until the final step. Rounding in the middle can shift the last digit of your answer.

Practice Problems: Find the Arc Length of a Circle

Try these five first.

1. A circle has a radius of $9$ cm and a central angle of $60^\circ$. Find the arc length.
2. A circle has a radius of $15$ m and a central angle of $120^\circ$. Find the arc length.
3. A circle has a diameter of $20$ cm and a central angle of $45^\circ$. Find the arc length.
4. A circle has a radius of $8$ cm and a central angle of $\frac{\pi}{4}$ radians. Find the arc length.
5. An arc is $18.85$ cm long on a circle with a radius of $12$ cm. Find the central angle in radians and in degrees.

Answer key:

1. $L = \frac{60}{360} \times 2\pi \times 9 = 3\pi \approx 9.42$ cm.
2. $L = \frac{120}{360} \times 2\pi \times 15 = 10\pi \approx 31.42$ m.
3. Radius $r = \frac{20}{2} = 10$ cm. $L = \frac{45}{360} \times 2\pi \times 10 = \frac{5\pi}{2} \approx 7.85$ cm.
4. $L = r\theta = 8 \times \frac{\pi}{4} = 2\pi \approx 6.28$ cm.
5. Rearrange $L = r\theta$: $\theta = \frac{L}{r} = \frac{18.85}{12} \approx 1.57$ radians, which is $\frac{\pi}{2}$ radians, or $90^\circ$. Check: $12 \times \frac{\pi}{2} = 6\pi \approx 18.85$ cm.

Frequently Asked Questions About Arc Length

Short answers.

1. What is the formula for the arc length of a circle? In degrees, $L = \frac{\theta}{360^\circ} \times 2\pi r$. In radians, $L = r\theta$, where $L$ is the arc length, $r$ is the radius, and $\theta$ is the central angle.

2. How do you calculate arc length step by step? Find the radius and angle, check the unit, pick the matching formula, then substitute and calculate.

3. Can you find arc length using the diameter? Yes. Halve the diameter to get the radius, $r = \frac{d}{2}$, then use the usual formula.

4. What is the difference between arc length and circumference? Circumference is the distance all the way round the circle, $2\pi r$. Arc length is the distance along part of the circle only.

5. How do you calculate arc length using radians? Use $L = r\theta$ with the angle in radians: multiply the radius by the angle, with no $360$ in the formula.

6. Is arc length the same as chord length? No. Arc length follows the curve, while the chord is the straight line between the same two points, so the arc is always longer.

7. How do you find the length of a major arc? Subtract the minor arc length from the full circumference: $L_{\text{major}} = 2\pi r - L_{\text{minor}}$. Or use the major angle, $360^\circ$ minus the minor angle, in the degree formula.

8. Should arc length answers be written in terms of pi? Exact answers stay in terms of $\pi$, such as $7\pi/2$. Give a decimal too when asked, rounding only at the last step.

Further Reading

These pages explain the same ideas from other angles.

• Maths is Fun: Circle Sector and Segment: arc length in degrees and radians, with interactive diagrams.
• Khan Academy: Arc length from angle measure: worked video examples of finding arc length.
• Wikipedia: Arc length: the general definition and the circle formulas.
• Wolfram MathWorld: Arc Length: the formal definition and the circle case.