Why must the angle be in radians and not degrees?

The formula s = r\theta relies on a specific mathematical relationship between angle and arc length that only holds when \theta is in radians. If you use degrees, the relationship breaks and you get an answer that is too large by a factor of about 57.3. Radians and degrees measure the same angle differently, and the formula was designed around radians.

How is arc length different from circumference?

Circumference is the total distance all the way around a circle (using the full angle of 2\pi radians). Arc length is the distance along just a portion of the circle for any angle \theta you choose. If your angle is 2\pi, then arc length equals circumference.

What if I only know the diameter, not the radius?

Divide the diameter by 2 to get the radius first, then use s = r\theta with that radius value. The arc length formula is built around radius, not diameter.

Does this formula work for very small or very large angles?

Yes, the formula works for any angle in radians. If the angle is very small (like 0.01 radians), the arc is short and nearly straight. If the angle is close to 2\pi radians (nearly a complete circle), the arc length approaches the full circumference 2\pi r.

Arc Length

Arc Length tells you the distance along a circular curve between two points, defined by the radius and the central angle in radians.

s = r\theta \quad (\theta \text{ in radians})

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What each symbol means

What Arc Length takes

Symbol	Meaning
$s$	The arc length is the distance measured along the curve of the circle, in the same units as the radius (such as centimeters, meters, or inches).
$r$	The radius is the distance from the center of the circle to any point on its edge, and must be in the same units as $s$ ; using the diameter instead will give an answer that is too large by a factor of 2.
$\theta$	The central angle is the angle at the center of the circle between the two radii that bound the arc, and must be measured in radians; if you use degrees instead, your answer will be wrong by a factor of about $\frac{180}{\pi} \approx 57.3$ .

When to use it

Use arc length when you need to find the distance along part of a circle's edge.

Level

Usually taught in: Geometry

Worked examples

1. Find arc length with a simple angle

Problem

A circle has radius 3 cm. Find the arc length when the central angle is

\frac{\pi}{6}

radians.

$s = r\theta$
We start with the arc length formula, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians.
$s = 3 \cdot \frac{\pi}{6}$
We substitute the values: $r = 3$ cm and $\theta = \frac{\pi}{6}$ radians.
$s = \frac{3\pi}{6} = \frac{\pi}{2}$
We simplify by dividing both numerator and denominator by 3, giving $s = \frac{\pi}{2}$ cm.

Answer: $s = \frac{\pi}{2} \text{ cm} \approx 1.57 \text{ cm}$

This example shows the basic use of the formula with a common angle. The angle $\frac{\pi}{6}$ is one-twelfth of a full rotation, so the arc length is one-twelfth of the full circumference $2\pi r = 6\pi$ cm.

2. Find arc length with a decimal radius and angle

Problem

A circle has radius 1.8 m. Find the arc length when the central angle is 2.5 radians.

$s = r\theta$
We apply the arc length formula.
$s = 1.8 \cdot 2.5$
We substitute $r = 1.8$ m and $\theta = 2.5$ radians.
$s = 4.5$
We multiply: $1.8 \times 2.5 = 4.5$ m.

Answer: $s = 4.5 \text{ m}$

When both the radius and angle are given as decimals, multiplication is straightforward. The angle 2.5 radians is about 143°, or roughly $\frac{2.5}{2\pi} \approx 40\%$ of a full circle, so the arc length is about 40% of the circumference.

3. Fencing a curved section of a circular garden

Problem

A circular herb garden has a radius of 5 meters. The owner wants to install decorative fencing along an arc between two points. The arc subtends a central angle of

\frac{3\pi}{4}

radians at the center. How many meters of fencing are needed?

$r = 5 \text{ m}, \quad \theta = \frac{3\pi}{4} \text{ radians}$
We identify the given values: radius $r = 5$ m and central angle $\theta = \frac{3\pi}{4}$ radians.
$s = r\theta = 5 \cdot \frac{3\pi}{4}$
We apply the arc length formula by substituting the known values into $s = r\theta$ .
$s = \frac{15\pi}{4} \text{ m}$
We multiply: $5 \times \frac{3\pi}{4} = \frac{15\pi}{4}$ m.
$s \approx \frac{15 \times 3.14159}{4} \approx 11.78 \text{ m}$
We approximate numerically by evaluating $\frac{15\pi}{4}$ to find the fencing length in meters.

Answer: $s = \frac{15\pi}{4} \text{ m} \approx 11.78 \text{ m}$

This practical problem shows why arc length matters in real contexts. Once the owner knows the arc length, they can purchase exactly the right amount of fencing material without waste. The angle $\frac{3\pi}{4}$ radians equals 135°, or three-eighths of a full circle.

Common mistakes

Where Arc Length usually goes wrong

The mistake	Why it is wrong	The fix
Writing $s = 5 \cdot 60°$ and getting $s = 300°$ when the radius is 5 and the angle is 60°.	The formula requires the angle to be in radians, not degrees; inserting degrees directly produces a unitless nonsense answer.	Convert the angle to radians first: $60° = \frac{\pi}{3}$ radians, then $s = 5 \cdot \frac{\pi}{3} = \frac{5\pi}{3}$ m.
Using $s = 2\pi r$ (the circumference formula) instead of $s = r\theta$ when you only need part of the circle.	Circumference gives the total distance around the entire circle, but arc length is only for a portion of it.	Use $s = r\theta$ where $\theta$ is the specific central angle of the arc you want, not $2\pi$ .
Substituting the diameter instead of the radius, writing $s = d \cdot \theta$ where $d$ is the diameter.	The formula uses radius, not diameter; using diameter will double your answer.	Always use the radius $r$ . If you are only given the diameter, divide it by 2 first to get the radius.

Tips and when to use something else

Always check that your angle is in radians—if the problem gives degrees, convert first using $\text{radians} = \text{degrees} \times \frac{\pi}{180}$ .
If you need the entire distance around the circle, use the Circumference of a Circle formula $C = 2\pi r$ instead; arc length applies only to part of the circle.
Arc length is always measured in the same units as the radius—if the radius is in meters, the arc length is in meters.
Check your work by estimating: the arc length should always be less than the full circumference $2\pi r$ (unless $\theta = 2\pi$ exactly).

Related formulas

Circumference of a CircleThe circumference of a circle is the total distance around its perimeter, calculated using either the radius or diameter.Area of a SectorCalculates the area of a circular sector enclosed by two radii and an arc when you know the radius and the central angle in radians.RatioA ratio compares two quantities to show their relationship, and can be written as a fraction or with a colon to help you scale or find missing values.

Frequently asked questions

Why must the angle be in radians and not degrees?: The formula $s = r\theta$ relies on a specific mathematical relationship between angle and arc length that only holds when $\theta$ is in radians. If you use degrees, the relationship breaks and you get an answer that is too large by a factor of about 57.3. Radians and degrees measure the same angle differently, and the formula was designed around radians.
How is arc length different from circumference?: Circumference is the total distance all the way around a circle (using the full angle of $2\pi$ radians). Arc length is the distance along just a portion of the circle for any angle $\theta$ you choose. If your angle is $2\pi$ , then arc length equals circumference.
What if I only know the diameter, not the radius?: Divide the diameter by 2 to get the radius first, then use $s = r\theta$ with that radius value. The arc length formula is built around radius, not diameter.
Does this formula work for very small or very large angles?: Yes, the formula works for any angle in radians. If the angle is very small (like 0.01 radians), the arc is short and nearly straight. If the angle is close to $2\pi$ radians (nearly a complete circle), the arc length approaches the full circumference $2\pi r$ .

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Reviewed 2026-09-18