The Distance You Cannot Measure with a Straight Ruler
Walk around the curved edge of a running track from one mark to another, and the distance you cover is not the straight line between the two marks; it is the curve itself. A ruler laid flat would undershoot it every time. That curved distance is the arc length, and the surprising part is that a single fraction of the full circle is all you need to find it exactly.
Once you see the arc as a slice of the whole way round, the formula stops being something to memorise and becomes a fraction you can rebuild any time.
What Is Arc Length?
Arc length is the distance along the curved boundary of a circle between two points, measured along the curve itself, not in a straight line. An arc is that curved piece of the circumference; the arc length is how long it is.
Because an arc is part of the circumference, its length is always a fraction of the whole way around the circle. The size of that fraction is set by the central angle, the angle the arc makes at the centre, written θ. A small angle gives a short arc; a $360°$ angle gives the entire circumference. The shorter arc between two points is the minor arc; the longer one is the major arc, and together they make the full circumference.
The Arc Length Formula — Degrees and Radians
The arc is a fraction of the circumference, and the central angle tells you which fraction. The full circumference is $2\pi r$, and an arc with central angle θ (in degrees) is $\dfrac{\theta}{360°}$ of it. Define the variables first: L is the arc length, r the radius, θ the central angle, and π ≈ 3.14159.
In degrees:
$$L = \frac{\theta}{360°} \times 2\pi r.$$
In radians. A radian is defined so that an angle of one radian cuts off an arc exactly equal to the radius. A full circle is $2\pi$ radians, so the fraction $\dfrac{\theta}{360°}$ becomes $\dfrac{\theta}{2\pi}$, and the $2\pi r$ cancels beautifully:
$$L = \frac{\theta}{2\pi} \times 2\pi r = r\theta \quad (\theta \text{ in radians}).$$
The radian formula $L = r\theta$ is the cleanest in all of geometry, and it is the whole reason radians exist. If your angle is in degrees and you want to use it, convert first: radians $= \text{degrees} \times \dfrac{\pi}{180}$.
[IMAGE PROMPT: Two side-by-side circle diagrams. Left: centre O, radius r, central angle theta in degrees, arc highlighted, formula below "L = (theta / 360) times 2 pi r". Right: centre O, radius r, central angle of exactly 1 radian marked, with the arc it cuts off drawn equal in length to the radius (both marked with a single tick), formula below "L = r theta (theta in radians)". Both diagrams clean, formulas fully visible, the radian diagram emphasising arc = radius when theta = 1.
Alt text: Two arc-length diagrams, one in degrees giving L = theta over 360 times 2 pi r, the other showing one radian cutting off an arc equal to the radius, giving L = r theta.]
How Do You Find Arc Length?
The usual case gives you the radius and the central angle: pick the formula matching the angle's unit and substitute. If the angle is in degrees, use $\dfrac{\theta}{360°}\times 2\pi r$; if in radians, use $r\theta$.
Two harder cases come up in exams. If you are given the radius and the sector area instead of the angle, the relationship Area $= \tfrac{1}{2}Lr$ lets you solve for L directly, no angle needed (this links arc length to the sector of a circle). And if you are given a chord length instead of the angle, you first recover the angle from the chord, $\text{chord} = 2r\sin!\left(\tfrac{\theta}{2}\right)$, then use the arc-length formula. The first of these we work below.
Examples of Arc Length
With the fraction idea and both formulas in hand, here is arc length being found in the cases a problem actually presents, from a clean degree calculation up to finding the arc from a sector area. The problems build from degrees to radians to a real-world track.
Example 1: Find the length of an arc that subtends a central angle of $90°$ in a circle of radius 14 cm. Use $\pi = \dfrac{22}{7}$.
The arc is $\dfrac{90}{360} = \dfrac{1}{4}$ of the circumference:
$$L = \frac{\theta}{360°} \times 2\pi r = \frac{90}{360} \times 2 \times \frac{22}{7} \times 14 = \frac{1}{4} \times 88 = 22 \text{ cm}.$$
Final answer: L = 22 cm.
Example 2: Find the arc length for a central angle of $60°$ in a circle of radius 9 cm.
A first instinct is to use the clean radian formula $L = r\theta$ and write $L = 9 \times 60 = 540$ cm. Check that against the picture. The whole circumference here is $2\pi r \approx 56.5$ cm, so no arc on this circle can be longer than that, yet 540 cm is almost ten times the full way round. The error: $r\theta$ only works when θ is in radians, and $60$ here means degrees.
Convert the angle to radians first, then apply $L = r\theta$:
$$\theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \text{ rad}, \qquad L = r\theta = 9 \times \frac{\pi}{3} = 3\pi \approx 9.42 \text{ cm}.$$
(Or use the degree formula directly: $L = \tfrac{60}{360}\times 2\pi(9) = 3\pi$ cm — the same answer.) Final answer: L = 3π ≈ 9.42 cm.
Example 3: A circle has a radius of 5 m. Find the length of an arc that subtends a central angle of 2 radians.
The angle is already in radians, so use $L = r\theta$ directly:
$$L = r\theta = 5 \times 2 = 10 \text{ m}.$$
Final answer: L = 10 m.
Example 4: Find the arc length cut off by a central angle of $\dfrac{\pi}{4}$ radians in a circle of radius 12 cm.
$$L = r\theta = 12 \times \frac{\pi}{4} = 3\pi \approx 9.42 \text{ cm}.$$
Final answer: L = 3π ≈ 9.42 cm.
Example 5: A sector of a circle has radius 8 cm and area 40 cm². Find the arc length of the sector.
Use the relationship between sector area and arc length, Area $= \tfrac{1}{2}Lr$, and solve for L:
$$40 = \frac{1}{2} \times L \times 8 ;\Rightarrow; 40 = 4L ;\Rightarrow; L = 10 \text{ cm}.$$
Final answer: L = 10 cm.
Example 6: A runner travels along the curved end of a circular athletics track of radius 36.5 m through a central angle of $180°$. How far does the runner travel? Use $\pi = 3.14$.
A $180°$ arc is half the circumference:
$$L = \frac{180}{360} \times 2 \times 3.14 \times 36.5 = \frac{1}{2} \times 229.22 = 114.61 \text{ m}.$$
Final answer: about 114.6 m.
Where Arc Length Shows Up
Arc length matters because so many real distances are measured along curves, not straight lines, and a straight ruler simply cannot read them.
Running tracks and roads. The lanes of an athletics track are curved, and each outer lane has a longer arc than the inner one, which is exactly why staggered starting lines exist: they equalise the arc length each runner covers.
GPS and the curved Earth. The distance between two cities along the Earth's surface is an arc on a great circle, not the straight chord through the planet. Navigation systems compute arc lengths to give real travel distances.
Gears, belts, and pulleys. The length of belt wrapped around a pulley is an arc; engineers size belts and chains by adding the straight runs to the arcs at each wheel.
Architecture and manufacturing. Cutting a curved moulding, bending a pipe, or laying a curved rail all need the true curved length, the arc, not the straight distance across.
For a Grade 10 student, arc length is where the radian quietly earns its place: the formula $L = r\theta$ is so clean only because radians were defined to make it so, and that single idea carries straight into trigonometry, circular motion, and the calculus definition of arc length for any curve.
Where Students Trip Up on Arc Length
Mistake 1: Using $L = r\theta$ with the angle in degrees
Where it slips in: A problem gives the angle in degrees, and the student plugs it straight into the radian formula.
Don't do this: Compute $L = r \times 60$ when $60$ means degrees.
The correct way: $L = r\theta$ only holds when θ is in radians. Either convert the angle ($\text{radians} = \text{degrees}\times\tfrac{\pi}{180}$) or use the degree formula $\dfrac{\theta}{360°}\times 2\pi r$. The rusher who grabs the cleanest formula without checking the unit is the one this catches.
Mistake 2: Confusing arc length with the chord
Where it slips in: A student measures or computes the straight distance between the arc's endpoints instead of the curved distance.
Don't do this: Treat the chord across the arc as the arc length.
The correct way: The arc length is measured along the curve and is always longer than the straight chord between the same two points. Use the arc formula, not the chord formula, when the question asks "how far along the edge."
Mistake 3: Forgetting to convert the final answer or mixing units
Where it slips in: A student uses a radius in metres and an arc-length value in centimetres in the same problem, or leaves an answer in $\pi$ when a decimal was asked for.
Don't do this: Mix metres and centimetres, or stop at $3\pi$ when the question wants a number.
The correct way: Keep all lengths in one unit throughout, and convert the final answer to the form the question requests. The silent understander who can do the maths still loses the mark if the unit is wrong.
Key Takeaways
Arc length is the distance along a circle's curved edge, a fraction of the circumference set by the central angle.
In degrees the formula is $L = \dfrac{\theta}{360°}\times 2\pi r$; in radians it simplifies to $L = r\theta$.
The radian formula is clean only because a radian is defined as the angle whose arc equals the radius.
With no angle given, use the sector-area relation Area $= \tfrac{1}{2}Lr$, or recover the angle from a chord first.
The most common mistake is using $L = r\theta$ with a degree angle — always check the unit, or that the arc doesn't exceed the circumference.
Practice These Problems to Solidify Your Understanding
Find the arc length for a central angle of $120°$ in a circle of radius 21 cm (use $\pi = \tfrac{22}{7}$).
Find the arc length for a central angle of $\dfrac{\pi}{6}$ radians in a circle of radius 18 cm.
A sector of radius 10 cm has area 60 cm². Find its arc length.
Answer to Question 1: 44 cm. Answer to Question 2: 3π ≈ 9.42 cm. Answer to Question 3: 12 cm. If Question 1 gave you a number larger than the circumference (~132 cm), you likely used $r\theta$ with degrees (see Mistake 1).
Want a live Bhanzu trainer to walk your child through arcs, sectors, and radians? Book a free demo class — online globally.
Was this article helpful?
Your feedback helps us write better content