How To Find The Length Of A Major Arc

You're staring at a circle problem. Which means the diagram shows a central angle, maybe 120 degrees, and the question asks for the length of the major* arc. Your brain instantly grabs the formula for arc length, plugs in 120, and moves on.

Wrong arc.

That 120-degree angle? It cuts out the minor* arc. In real terms, the major arc is the long way around — the rest of the circle. If you only calculate the short path, you've answered a question nobody asked.

Let's fix that.

What Is a Major Arc

Every circle has two arcs between any two points on its circumference. Now, the major arc is the longer one. Also, the minor arc is the shorter path. Simple enough — until you're under time pressure and your muscle memory takes over.

A major arc measures more than 180 degrees. The minor arc's central angle is less than 180°. Its central angle is greater than 180° (or π radians). Together, they always sum to 360° (or 2π radians).

You'll see major arcs named with three letters — like arc ACB — to distinguish them from the minor arc AB. Plus, that middle letter? It's a point on the major arc itself, just to make the path unambiguous.

The Relationship You Can't Ignore

Here's the thing most textbooks rush past: the major arc and minor arc are partners. They share endpoints. On top of that, they share the same radius. And their degree measures always add up to 360°.

If the minor arc is 120°, the major arc is 240°. On the flip side, if the minor arc is 85°, the major arc is 275°. Even so, this relationship is your safety net. Every single time.

Why It Matters

You might wonder — does anyone actually use major arc length outside of geometry class?

Surveyors do. When mapping property boundaries that curve around a hill or follow a circular easement, the "long way around" is often the legal boundary. So civil engineers calculating materials for curved roads or pipelines need the full arc length, not just the chord or the minor arc. CNC programmers cutting circular profiles on a lathe or mill — if the toolpath follows the major arc, the code needs that exact length for feed rate and timing calculations.

Even in everyday stuff: a circular running track, a round garden bed edging, the crust on a pizza slice where you took the big piece. The major arc shows up more than you'd think.

And on standardized tests? It's a classic trap. The problem gives you the minor central angle. The question asks for the major arc length. Students who don't pause lose points they knew how to earn.

How to Find the Length of a Major Arc

There are two main paths. Both start the same way: find the circle's circumference. Then take the right fraction of it.

Method 1: Subtract the Minor Arc from the Circumference

This is the most foolproof approach. You calculate the full circle, calculate the minor arc, subtract.

Step 1: Find the circumference.
C = 2πr (if you have the radius)
C = πd (if you have the diameter)

Step 2: Find the minor arc length.
If the central angle θ is in degrees:
Minor arc length = (θ/360) × C

If θ is in radians:
Minor arc length = θ × r

Step 3: Subtract.
Major arc length = C − Minor arc length

Let's run an example. That said, circle radius = 10 cm. Central angle of the minor arc = 100°.

Circumference = 2π(10) = 20π cm
Minor arc = (100/360) × 20π = (5/18) × 20π = (100π/18) = (50π/9) cm
Major arc = 20π − (50π/9) = (180π/9 − 50π/9) = (130π/9) cm ≈ 45.4 cm

Notice we kept π symbolic as long as possible. Rounding too early is where precision dies.

Method 2: Use the Major Central Angle Directly

If you know (or can find) the major arc's central angle, you can skip the subtraction.

Major central angle = 360° − minor central angle
Then: Major arc length = (Major angle / 360) × C

Same example: minor angle = 100°, so major angle = 260°.
Major arc = (260/360) × 20π = (13/18) × 20π = (260π/18) = (130π/9) cm

Same answer. Method 2 is faster if you're confident about the major angle. Method 1 is safer because the minor angle is usually what's given — you're less likely to misread the diagram.

Working in Radians

Radians make the arithmetic cleaner, once you're used to them.

Circumference = 2πr
Minor arc length = θ_minor × r
Major arc length = (2π − θ_minor) × r

That's it. That said, the major arc's radian measure is just 2π minus the minor angle. Worth adding: multiply by radius. Done.

For more on this topic, read our article on what is equal to 4 5 or check out how to find x intercepts of a quadratic.

Example: r = 6 m, minor angle = π/3 radians.
Major arc = (2π − π/3) × 6 = (5π/3) × 6 = 10π m ≈ 31.4 m

No fraction-of-circumference step needed. Radians were made* for arc length.

When You're Given the Chord Length Instead

Sometimes the problem gives you the chord length (the straight line between the arc's endpoints) and the radius, but not the angle. You can still find the major arc length — it just takes an extra step.

Chord length c = 2r sin(θ/2) where θ is the minor* central angle in radians.

Solve for θ:
θ = 2 arcsin(c / 2r)

Then major arc length = (2π − θ) × r

This comes up in engineering and machining more than in textbooks. If you're writing G-code for a CNC machine and the drawing only shows chord length and radius, this is how you get the major arc feed distance.

Common Mistakes

Using the Wrong Angle

The number one error. On top of that, fix: Always circle the phrase "major arc" in the problem statement. Literally circle it with your pen. Which means the diagram shows a 70° angle. You've just calculated the minor arc. The problem asked for the major arc.
That said, you plug 70 into the formula. Force your brain to register: I need the big one.

Forgetting to Convert Degrees to Radians (or Vice Versa)

The formula s = rθ only works if θ is in radians*. If you plug in degrees, your answer will be off by a factor of 180/π ≈ 57.But 3. That's not a rounding error — that's a completely different number.

Forgetting to Convert Degrees to Radians (or Vice Versa)

The formula s = rθ only works if θ is in radians*. But if you plug in degrees, your answer will be off by a factor of 180/π ≈ 57. 3. So naturally, that's not a rounding error — that's a completely different number. If you're using the fraction-of-circumference method (θ/360) × C, then you must* use degrees. And mixing units here is like trying to add apples and engine oil — the result is meaningless. Fix: Write "degrees" or "radians" next to every angle value as you read the problem. Don't trust your memory.

Rounding π Too Early

We've said it before: don't replace π with 3.If you're computing (130π/9) cm, calculate 130/9 first, then multiply by π.
2/9 ≈ 45.39... That said, ≈ 45. 14 at the start: (130 × 3.14.14)/9 = 408.14 until the very last step. 14159... In practice, 36 cm
Close, but not the same. cm
If you'd rounded π to 3.130 ÷ 9 = 14.444... 444...
Because of that, × 3. In precision work, that difference matters.

Confusing Arc Length with Chord Length

Arc length is the distance along the curve*. Chord length is the straight-line distance between the two endpoints. They are not the same (except in the limit as the angle approaches zero).
Also, if a problem says "the arc is 10 cm long," that's arc length. If it says "the distance between the endpoints is 10 cm," that's chord length — and you'll need to work backwards to find the angle first.

Sign Errors with Radians

When working in radians, remember that 2π radians = 360°. So if your minor angle is θ, your major angle is (2π − θ). Don't accidentally write (θ − 2π) — that gives you a negative number, which doesn't make sense for a physical length.
If you get a negative arc length, check your angle subtraction.

Quick Reference: Which Method to Use

Final Thoughts

Finding the length of a major arc isn't about memorizing a special formula — it's about understanding that the major arc is simply what remains when you take the major central angle's share of the full circumference. Whether you work in degrees or radians, whether you subtract from the total or calculate directly, the underlying principle is the same: the major arc is the long way around the circle.

The key is staying organized. Because of that, label your angles clearly. Keep π symbolic as long as you can. Double-check that you're solving for the right arc. And always, always verify that your final answer makes sense — if the major arc length is shorter than the minor arc, something has gone wrong.

In practical applications — surveying, machining, architecture — these calculations determine real distances, real materials, real costs. Getting them right isn't just about passing a test. It's about building things that work.

So the next time you see a circle with an arc labeled "major," remember: it's not just the bigger arc. It's the one that goes the long way around, and now you have every tool you need to measure it precisely.