Arc Length Calculator

The DigiCalc arc length calculator measures the exact distance along a curved section of a circle. It needs just two inputs: the radius and the central angle. You enter your values, and the tool returns the arc length, the sector area, and the chord length in one step. There is no need to work through the arc length formula by hand.

This tool helps students checking geometry homework, engineers laying out a curved road, and designers cutting a circular pattern. It removes the manual arithmetic and the rounding mistakes that come with it. Arc length appears everywhere a straight ruler cannot reach. Think of a running track curve, the sweep of a clock hand, the bend of a pipe, or the edge of a pizza slice.

This page explains how to find arc length in both radians and degrees. It shows the full arc length equation with worked examples. It also extends into the calculus method for the arc length of any curve, not just a circle.

What Is Arc Length?

Arc length is the distance measured along the curved boundary of a circle between two points. It follows the curve rather than cutting straight across it. Arc length is a portion of the circle's total circumference. The size of the central angle sets how much of the circumference the arc covers. The larger the central angle, the longer the arc.

A full circle has a central angle of 360 degrees, or 2 times pi radians. Its arc length equals the whole circumference, 2 times pi times the radius. A half circle has an arc length of exactly half the circumference. Arc length scales directly with the angle, so it is a simple proportion. The arc is to the circumference as the central angle is to a full turn.

This proportional relationship is the key insight behind every arc length problem. A 60 degree slice covers one sixth of the 360 degree circle, so its arc covers one sixth of the circumference. You can solve many arc length questions with this ratio alone. Just ask what fraction of the full circle your angle represents.

Arc Length vs Circumference and Chord

It helps to keep three related measurements clear. The circumference is the full distance around the entire circle. The arc length is the distance around only the part marked off by your central angle. The chord is the straight shortcut between the two ends of that arc. For any angle below 360 degrees, the order from longest to shortest is always circumference, then arc, then chord.

For a 90 degree angle on a 10 cm radius, the arc length is 15.71 cm. The chord is only 14.14 cm. The gap between them grows as the angle widens. That is why a curved path is always longer than the straight line between its endpoints.

Arc Length Formula and Equation

The arc length formula depends on whether the central angle is in radians or degrees. The radian form is the cleanest version of the arc length equation:

Arc length (radians): L = r × θ

Here L is the arc length, r is the radius, and θ (theta) is the central angle in radians. This compact arc of a circle formula works because of how the radian is defined. One radian is the angle that creates an arc exactly equal to the radius. The radian is the coherent SI unit of plane angle, according to the International Bureau of Weights and Measures (BIPM). That is why the formula needs no extra conversion factor.

For example, take a circle with a radius of 10 cm and a central angle of 2 radians. Its arc length is 10 × 2 = 20 cm. The arc length equation is that direct. Multiply the radius by the angle in radians, and you have the answer.

Why Arc Length Uses Radians

The radian exists to make the arc of a circle formula as simple as possible. One radian is the central angle that cuts off an arc as long as the radius. So an angle in radians already measures how many radius lengths fit along the arc. Multiply by the radius, and you convert that count into a real distance.

Degrees, by contrast, are an arbitrary historical choice that splits the circle into 360 parts. The degree formula needs an extra factor of pi over 180 to reach the natural radian measure. A full turn holds 2 times pi radians, about 6.283. The radian form L = r × θ falls straight out of that definition. For serious geometry or calculus work, radians are the standard unit of angle.

How to Find the Arc Length of a Circle

To find the arc length of a circle, you need two values: the radius and the central angle. Follow these steps:

1. Identify the radius (r), the distance from the center of the circle to its edge.
2. Identify the central angle (θ) that opens across the arc you want to measure.
3. If the angle is in degrees, use the degree formula below or convert it to radians first.
4. Multiply the radius by the angle in radians: L = r × θ.

That is how to find arc length for any circular arc. The DigiCalc arc length calculator runs these steps automatically. It lets you switch between radians and degrees, so you never convert by hand.

Arc Length Formula in Degrees

When the central angle is in degrees, use the arc length formula degrees version. The equation is L = (θ / 360) × 2 × pi × r. You take the fraction of the full circle your angle covers. Then you multiply by the full circumference. For a 90 degree angle on a radius of 6 cm, L = (90 / 360) × 2 × 3.14159 × 6 = 9.42 cm.

Arc Length Formula in Radians

The radian version of the arc length formula is L = r × θ. Here the angle is expressed in radians. Mathematicians and engineers prefer this form because it avoids the 360 and pi factors. A quarter circle has an angle of pi / 2 radians. A half circle has pi radians. A full circle has 2 times pi radians.

Arc Length Conversion Table

This table lists common central angles and the arc length they produce for several radii. Use it as a quick reference. You can also enter your own values into the DigiCalc arc length calculator for an exact result.

How to Use This Arc Length Solver

The DigiCalc arc length solver is built for speed and accuracy. To calculate arc length:

1. Enter the radius of your circle.
2. Enter the central angle and choose degrees or radians.
3. Read the arc length result, shown instantly with the sector area and chord length.
4. Adjust any value to see the result update in real time.

The tool accepts both angle units. So it doubles as a quick way to check work in a classroom or on a job site. You need no separate conversion step.

Sector Area Calculator: Area of the Sector Formula

A sector is the pie-slice region bounded by two radii and the arc between them. The area of the sector formula in radians is A = (1 / 2) × r² × θ. In degrees it is A = (θ / 360) × pi × r². The DigiCalc tool doubles as a sector area calculator. It returns the sector area alongside the arc length, so you do not need a second tool.

Chord Length Calculator

A chord is the straight line connecting the two endpoints of an arc. It cuts across the circle instead of following the curve. The chord length calculator uses the formula chord = 2 × r × sin(θ / 2), where θ is the central angle. The chord is always shorter than the arc it spans. A straight path is the shortest distance between two endpoints.

Chord length matters in construction. A curved arch is often specified by the straight span across its base plus the rise at its center. Knowing the chord and the radius lets you recover the central angle. From there, you can find the arc length of the curved member itself.

Central Angle Calculator

If you know the arc length and the radius, you can work backward to find the central angle. The central angle calculator rearranges the main formula to θ = L / r. This gives the angle in radians. Take an arc length of 15 cm on a radius of 10 cm. The central angle is 15 / 10 = 1.5 radians, about 85.9 degrees. This reverse calculation helps when you measure a physical arc and need its angle.

Arc Radius Calculator: Find the Radius from Arc Length

You can also rearrange the formula to work as an arc radius calculator. When you know the arc length and the central angle in radians, the radius is r = L / θ. For an arc length of 20 cm spanning 2 radians, the radius is 20 / 2 = 10 cm. This helps when you can measure a curved edge and its angle but cannot reach the center. The arc radius value then unlocks every other property, including the circumference and sector area.

Arc Length Integral and Curve Length Calculator

For curves that are not part of a circle, the simple L = r × θ formula no longer applies. You then need the arc length integral from calculus. The formula is L = the integral of the square root of (1 + (dy/dx)²), taken with respect to x between the two endpoints. This is the arc length formula calculus relies on for any smooth curve. This curve length formula sums up countless tiny straight segments along the curve.

The same arc length by integration idea extends to other coordinate systems. For a parametric curve with x(t) and y(t), the arc length of a parametric curve uses a related form. It integrates the square root of ((dx/dt)² + (dy/dt)²) dt. For a polar curve r(θ), the arc length of polar curve uses a form that integrates the square root of (r² + (dr/dθ)²) dθ. These arc length integral formula versions, also called the arc length formula integral, handle any smooth curve. Arc length parameterization then rewrites a curve so distance along it becomes the variable itself.

The DigiCalc tool focuses on circular arcs. Still, the arc length calculus background shows why the circle case is the simplest member of a larger family. The arc length of a curve always comes from adding up infinitesimally short straight pieces. This holds whether the shape is a circle, a parabola, or a spiral. It is the same idea any arc length curve calculator or curve length calculator uses behind the scenes.

As a worked example, take the curve y = x squared from x = 0 to x = 1. Its derivative is dy/dx = 2x. So the arc length integral becomes the integral from 0 to 1 of the square root of (1 + 4x squared) dx. Evaluating this gives an arc length of about 1.479 units. A straight line between the same endpoints would measure only 1.414 units. This confirms that the curved path is longer, exactly as expected.

Common Mistakes When Finding Arc Length

The most frequent error is mixing up angle units. The formula L = r × θ only works when the angle is in radians. Plugging a degree value straight in gives a result that is far too large. Always convert to radians first, or switch the DigiCalc tool to degree mode. Degree mode handles the conversion for you.

A second common mistake is confusing arc length with chord length. The arc follows the curve, while the chord cuts straight across. The two values are never equal for a real angle. A third pitfall is using the diameter in place of the radius. The diameter is twice the radius, so this doubles every answer. Check that your radius is the center-to-edge distance to avoid this. Remember too that a 360 degree arc length equals the full circumference, a useful sanity check.

Arc Length Examples

Worked examples make the arc of a circle formula concrete. Each example below uses real numbers. You can follow the calculation end to end and find the arc length yourself.

Example 1, running track: A curved lane has a radius of 36.5 meters and bends through 180 degrees. Arc length = (180 / 360) × 2 × 3.14159 × 36.5 = 114.67 meters. That is the distance a runner covers around the bend.

Example 2, clock hand: The tip of a 15 cm minute hand sweeps 90 degrees in 15 minutes. Arc length = (90 / 360) × 2 × 3.14159 × 15 = 23.56 cm traveled by the tip.

Example 3, radians directly: A gear of radius 8 cm turns through 2.5 radians. Arc length = 8 × 2.5 = 20 cm of belt pulled across the gear.

Example 4, pizza crust: A pizza with a 14 cm radius is cut into 8 equal slices. Each slice has a central angle of 45 degrees. The crust on one slice is the arc length = (45 / 360) × 2 × 3.14159 × 14 = 10.996 cm. That is almost 11 cm of crust per slice.

Example 5, reverse with central angle: Suppose you measure an arc of 25 cm on a wheel of radius 20 cm. Using the central angle calculator method, the angle = 25 / 20 = 1.25 radians, about 71.6 degrees. This shows how arc length and angle are two sides of the same relationship.

Real-World Uses of Arc Length

Arc length is not just a textbook idea. Civil engineers calculate it to design curved highways, railway bends, and bridge arches. Each must follow a precise radius. Manufacturers use it to cut sheet metal, fabric, and curved trim to the exact length before bending. In sports, the staggered start lines on a running track exist because outer lanes have longer arc lengths around each curve.

Arc length also drives everyday technology. A robotic arm rotating a tool through a known angle travels a predictable arc. The controller uses L = r × θ to position it. Watchmakers, animators, and architects rely on the same arc of a circle formula. It translates an angle into a real, measurable distance along a curve.

The same principle scales up to astronomy and navigation. The Earth is roughly spherical. So the distance between two points along a line of latitude is an arc length. You find it by multiplying the Earth's radius by the angle between them in radians. A single degree of arc along the equator spans roughly 111 km. Surveyors, pilots, and satellite engineers all calculate arc length to plan routes that follow the curve of the planet.

Arc Length in Radians vs Degrees: Which to Use

Both angle units give the same arc length, so the choice depends on your context. Use degrees when your angle is already stated in degrees, such as a 45 degree slice or a 120 degree turn. The degree formula handles the conversion. Use radians for calculus, physics, or engineering. There, angles naturally appear in radians, and the clean L = r × θ form keeps the math simple.

The DigiCalc arc length calculator supports both, so you never have to commit to one. To move between them, multiply degrees by pi over 180 to get radians. Or multiply radians by 180 over pi to get degrees. For instance, 90 degrees equals pi over 2 radians, and 1 radian equals about 57.3 degrees.

Limitations of the Arc Length Calculator

This arc length calculator assumes a perfect circular arc with a constant radius. Real objects are rarely flawless circles. So a measured physical arc may differ slightly from the calculated value. The tool also treats the radius and angle as exact inputs. Any error in measuring those values carries straight through to the result.

For curves that are not circular, such as ellipses, parabolas, or spirals, the simple formula does not apply. You must use the arc length integral from calculus instead. Results are also rounded for display. For high-precision engineering work, carry extra decimal places through your own calculation. DigiCalc follows standard mathematical conventions consistent with the numerical computing standards published by the Institute of Electrical and Electronics Engineers (IEEE).

Related Calculators

Explore more circle and geometry tools from DigiCalc. Use the sector area calculator to find the area enclosed by an arc. Try the area of a circle calculator for the full circle. Switch angle units with the radians to degrees converter. Study a circle from its equation with the circle equation analyzer. For the full set of tools, visit the DigiCalc math calculators hub.