Find The Length Of The Bolded Arc.

You're staring at a circle. Part of its edge is highlighted — maybe shaded, maybe drawn thicker, maybe just labeled "arc AB" in a textbook diagram. The question asks for the length of that bolded arc. And you're thinking: wait, do I use the radius? Even so, the diameter? Here's the thing — the angle? Is it in degrees or radians?

Yeah. It's one of those geometry problems that looks simple until you actually have to solve it.

What Is Arc Length, Really

Arc length is just the distance along the curved edge of a circle between two points. Not the straight-line distance — that's a chord. Plus, the curved* distance. The part you'd trace with a pencil if you followed the circle's edge from point A to point B. And that's really what it comes down to.

Here's the thing most people forget: every arc is a fraction of the whole circumference. In practice, that's the entire conceptual key. Once you internalize that, the formulas stop being things to memorize and start being things you can derive on a napkin.

A full circle gives you 360° (or 2π radians) and a circumference of 2πr. An arc gives you some central angle θ and a piece of that circumference. The ratio stays the same:

arc length / circumference = central angle / full angle

That's it. That's the whole idea.

The Two Formulas You'll Actually Use

If your angle is in degrees:

s = (θ/360) × 2πr

If your angle is in radians:

s = θr

The radian version is cleaner. Practically speaking, that's not an opinion — radians are defined so that this works out neatly. So θ radians subtends θ times the radius. One radian subtends an arc length equal to the radius. Done.

But degrees show up constantly in textbooks and standardized tests, so you need both.

Why This Trips People Up

Three main reasons. Maybe four.

First: Confusing arc length with chord length. The chord is the straight line connecting the endpoints. The arc is the curve. They're different numbers unless the angle is tiny. I've seen students calculate the chord, write it down as the arc length, and move on. Don't be that student.

Second: Radians vs. degrees. If you plug 60 into s = θr thinking it's degrees, you'll get an answer 57 times too big. (Because 60° = π/3 radians ≈ 1.047 radians. Not 60.) Always check your angle unit. Always.

Third: Using the diameter instead of the radius. The formula uses r. The problem might give you d. Half it. Every time. This is the most common careless error I see — and I've made it myself more than once.

Fourth: Not realizing the angle must be the central* angle — the one with its vertex at the circle's center. An inscribed angle (vertex on the circle) is half the central angle that subtends the same arc. If the problem gives you an inscribed angle, double it before using the formula.

How to Solve Any Arc Length Problem

Let's walk through the process like you're sitting at a desk with a pencil.

Step 1: Identify What You're Given

Circle radius? Diameter? Central angle in degrees? In practice, in radians? On top of that, inscribed angle? Arc measure in degrees (which equals the central angle measure)?

Write it down. Worth adding: label it. Don't just hold it in your head.

Step 2: Convert to the Right Form

• Diameter given? Radius = d/2.
• Angle in degrees but you want the radian formula? Multiply by π/180.
• Angle in radians but the problem expects degrees? Multiply by 180/π.
• Inscribed angle given? Central angle = 2 × inscribed angle.

Step 3: Pick Your Formula

Degrees → s = (θ/360) × 2πr
Radians → s = θr

Both give the same answer. Use whichever feels faster with your numbers.

Step 4: Plug In and Simplify

This is where arithmetic kills you. That said, π stays as π unless the problem asks for a decimal approximation. If it says "round to the nearest tenth," then sure, use 3.14159... and round at the end. In practice, not in the middle. Rounding mid-calculation accumulates error.

Step 5: Sanity Check

Does your answer make sense?

• Arc length should be less than the full circumference (2πr).
• If θ = 180°, arc length should be exactly half the circumference (πr).
• If θ = 90°, arc length should be a quarter (πr/2).
• If θ = 360°, arc length = 2πr (the whole circle).

If your number violates these, something's wrong.

Worked Examples

Example 1: Clean Numbers, Degrees

Radius = 10 cm. Also, central angle = 72°. Find arc length.

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Degrees formula: s = (72/360) × 2π(10)
s = (1/5) × 20π
s = 4π cm ≈ 12.57 cm

Check: 72° is 1/5 of 360°. Circumference = 20π. 1/5 of that = 4π.

Example 2: Radians, Messy Angle

Radius = 6 m. Central angle = 5π/6 radians.

Radian formula: s = (5π/6) × 6 = 5π m ≈ 15.71 m

Check: 5π/6 radians = 150°. Also, that's 150/360 = 5/12 of the circle. This leads to circumference = 12π. 5/12 × 12π = 5π.

Example 3: Diameter Given, Inscribed Angle

Diameter = 14 in. Inscribed angle = 40°.

First: radius = 7 in.
Even so, central angle = 2 × 40° = 80°. Worth adding: s = (80/360) × 2π(7) = (2/9) × 14π = 28π/9 in ≈ 9. 77 in.

Notice the chain: diameter → radius, inscribed → central. Miss either conversion and you're wrong.

Example 4: Working Backward — Find the Angle

Arc length = 15π cm. Radius = 18 cm. Find the central angle in degrees.

Use s = (θ/360) × 2πr
15π = (θ/360) × 2π(18)
15π = (θ/360) × 36π
15 = (θ/360) × 36
15 × 360 = 36θ
5400 = 36θ
θ = 150°

Or use radians first: s = θr → 15π = θ(18) → θ = 15π/18 = 5π/6 radians → × 180/π = 150°. Same answer, sometimes faster.

Common Mistakes (And

Common Mistakes (And How to Avoid Them)

1. Confusing radius and diameter
This is the number one error. Always, always* convert diameter to radius as your very first step. Write "r = d/2" explicitly

2. Using the diameter in the formula
Even after finding the radius, muscle memory types 2πd or plugs d into s = θr. The formulas require* radius. No exceptions.

3. Mixing degrees and radians
Plugging a degree measure into s = θr gives an answer that's off by a factor of 180/π (≈ 57.3). Plugging radians into the degree formula gives a tiny, nonsensical fraction. Pick a lane and convert before* you substitute.

4. Forgetting the "2" in the degree formula
s = (θ/360) × πr² is the area* formula missing a division by 2. The circumference is 2πr. That "2" matters.

5. Rounding π too early
4π is an exact answer. 12.56 is an approximation. If you round π to 3.14 in step 2 of a 5-step problem, your final answer drifts. Keep π symbolic until the very last line.

6. Ignoring the "inscribed angle" trap
An inscribed angle is half* the central angle that subtends the same arc. If the problem gives you an angle with its vertex on the circle, double it before using any arc length formula.

7. Units mismatch
Radius in feet, answer expected in inches? Convert first. Arc length inherits the linear unit of the radius. If r = 3 m, s is in meters. Not centimeters, not feet.

When Arc Length Shows Up Elsewhere

You'll see this again. It's not just a geometry unit.

• Trigonometry: The definition of radian measure is θ = s/r. The unit circle (r = 1) makes arc length numerically equal to the angle in radians. This is why calculus uses radians—derivatives of sin and cos only work cleanly when θ = s/r.
• Physics: Angular velocity ω (rad/s) relates to linear velocity v by v = ωr. That's just s = θr differentiated with respect to time. Distance traveled along a circular path is arc length.
• Calculus: Arc length of any curve y = f(x) from a to b generalizes this idea: ∫√(1 + (dy/dx)²) dx. The circle formula is the special case where the derivative is constant.
• Navigation & Engineering: Great-circle distances on Earth are arc lengths on a sphere. Latitude/longitude differences give you the central angle; Earth's radius gives you the scale.

The Mental Model That Sticks

Stop memorizing two formulas. Memorize one relationship:

Arc length is a fraction of the circumference.

That fraction is θ/360° if you think in degrees, or θ/2π if you think in radians. Since circumference = 2πr:

• Degrees: s = (θ/360) × 2πr
• Radians: s = (θ/2π) × 2πr = θr

Same logic. Same result. One less thing to forget.

Next time you see a circle problem, draw the circle. Mark the radius. Shade the arc. Ask: "What fraction of the way around is this?" Then multiply that fraction by 2πr. The formula writes itself.