Arc Length Calculator

Calculating arc length comes down to figuring out what fraction of a circle's full circumference you're dealing with. A full circle has a circumference of 2πr, so if your arc spans only part of that circle, you multiply the full circumference by the fraction of the total angle your arc covers.

The central angle determines that fraction. If the angle is given in degrees, you divide it by 360. If it's in radians, you divide it by 2π. Either way, the logic is the same: arc length equals the fraction of the circle times the full circumference.

You need two pieces of information: the radius of the circle and the central angle subtended by the arc. With those two values, the calculation is straightforward.

There are two versions of the arc length formula depending on whether your angle is measured in degrees or radians.

• Degrees: Arc Length = (θ / 360) × 2πr
• Radians: Arc Length = r × θ

In both formulas, r is the radius and θ (theta) is the central angle. The radian version is notably simpler, which is one reason radians are preferred in higher-level math and physics. When the angle is in radians, arc length is just the radius multiplied by the angle. Clean and direct.

To calculate arc length, plug your known values directly into the appropriate formula. The key is making sure your angle is in the right unit before you start. Mixing degrees and radians in the same formula will give you a wrong answer every time.

Here's a quick example. Say you have a circle with a radius of 8 cm and a central angle of 45 degrees. Using the degree formula: Arc Length = (45 / 360) × 2π × 8 = 0.125 × 50.27 ≈ 6.28 cm.

If that same angle were given as π/4 radians instead, you'd use the radian formula: Arc Length = 8 × (π/4) ≈ 6.28 cm. Same answer, different path.

An arc and a sector are closely related. A sector is the pie-slice shaped region of a circle bounded by two radii and the arc between them. The arc is just the curved edge of that sector.

Because they share the same central angle and radius, the formulas for arc length and sector area are built from the same components. Understanding one makes the other easier to work with.

If you know the arc length and the central angle but not the radius, you can rearrange the arc length formula to solve for r.

• From radians: r = s / θ
• From degrees: r = (s × 360) / (2π × θ)

For example, if an arc has a length of 15 cm and the central angle is 3 radians, then r = 15 / 3 = 5 cm. If the angle were 120 degrees instead: r = (15 × 360) / (2π × 120) = 5400 / 753.98 ≈ 7.16 cm.

This rearrangement is useful in engineering and design contexts where you're working backwards from a measured curve to find the circle it belongs to.

Similarly, if you know the arc length and the radius, solving for the central angle is just a matter of isolating θ in the formula.

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

Say the arc length is 10 inches and the radius is 4 inches. In radians: θ = 10 / 4 = 2.5 radians. To convert that to degrees: 2.5 × (180 / π) ≈ 143.24 degrees.

This comes up often in navigation, mechanical design, and anywhere you need to figure out how much of a circle an arc spans given physical measurements.

Sometimes you'll need to convert between degrees and radians before applying the arc length formula. The conversion is simple:

• Degrees to radians: multiply by π/180
• Radians to degrees: multiply by 180/π

Here are a few worked examples:

Notice that a 180-degree arc (a semicircle) gives an arc length of πr, which is exactly half the circumference. That's a quick way to double-check your work on common angles.

These two measurements are easy to confuse. The arc length is the actual curved distance along the circle between two points. The chord length is the straight-line distance between those same two points, cutting directly through space.

For any arc less than a full circle, the chord is always shorter than the arc. The bigger the central angle, the more the two values diverge. At 180 degrees, the chord is the diameter (2r), while the arc length is πr, which is noticeably longer.

The chord length formula uses trigonometry:

Chord Length = 2r × sin(θ/2)

Where θ is the central angle. Compare that to Arc Length = rθ (in radians). For small angles, the two are very close in value. That's actually the basis of a common approximation in physics: for small angles, arc length and chord length are nearly equal, so sin(θ) ≈ θ when θ is small and measured in radians.

It helps to have the core circle formulas in one place. Here's a quick reference:

These formulas all stem from the same fundamental relationship between a circle's radius and its central angle. Once you're comfortable with circumference and arc length, the rest tend to follow naturally.

Arc length isn't just a classroom concept. It shows up in a surprising number of real-world situations.

• Road and highway design: Engineers use arc length to calculate curved road sections, on-ramps, and roundabouts. The radius of curvature and the sweep angle determine how long the curved portion of road actually is.
• Mechanical engineering: Gears, pulleys, and conveyor belts all involve arcs. Calculating how far a belt travels around a pulley requires arc length.
• Architecture: Curved walls, arched ceilings, and circular windows all require arc length calculations during the design and materials-estimation phase.
• Navigation: Great-circle routes on a globe are arcs. Pilots and ship navigators use arc length (scaled to Earth's radius) to calculate travel distances.
• Animation and graphics: When an object moves along a curved path in a game or animation, the speed and timing often depend on arc length calculations to keep motion smooth and realistic.

Even something as everyday as a clock uses arc length. The tip of the minute hand traces an arc every hour, and knowing the radius of the clock face tells you exactly how far it travels. The math is everywhere once you start looking for it.