Arc Length Calculator

Introduction

Arc length is the distance along the curved part of a circle between two points. Think of it like measuring a piece of string that follows the curve instead of cutting straight across. The arc length calculator on this page helps you find that curved distance quickly and easily. All you need to know is the radius of your circle and the angle of the arc, and the tool does the math for you. Whether you're working on a geometry homework problem, a design project, or just trying to understand circles better, this calculator gives you an accurate answer in seconds.

How to Use Our Arc Length Calculator

Enter the radius and central angle of a circle, and this calculator will find the arc length for you.

Radius: Type in the radius of your circle. This is the distance from the center of the circle to any point on its edge. You can use any unit of measurement, like inches, centimeters, or feet.

Central Angle: Enter the central angle that forms the arc. This is the angle at the center of the circle between two radius lines. You can enter the angle in degrees or radians, depending on the option you choose.

Angle Unit: Pick whether your central angle is in degrees or radians. Degrees go from 0 to 360, while radians go from 0 to 2π (about 6.28).

Result — Arc Length: After you enter your values, the calculator will show you the arc length. The arc length is the distance along the curved part of the circle between the two endpoints of the arc. It will be in the same unit you used for the radius.

Arc Length Calculator

Arc length is the distance along a curved line that forms part of a circle or any other curve. Think of it like this: if you could take a piece of string and lay it perfectly along a curve, then straighten it out, the length of that string would be the arc length. It is one of the most useful measurements in geometry and shows up in everything from building roads to designing roller coasters.

Arc Length of a Circle

When you are working with a circle, the arc length depends on two things: the radius of the circle and the central angle that the arc covers. The central angle is the angle formed at the center of the circle between two radii that touch the ends of the arc. The formula is simple:

Arc Length (L) = r × θ

Here, r is the radius and θ (theta) is the central angle measured in radians. If your angle is in degrees, you first convert it to radians by multiplying by π/180. For example, a 90° angle on a circle with a radius of 5 meters gives an arc length of 5 × (π/2) ≈ 7.854 meters. If you need to find the total area enclosed by a circle rather than the arc, you can use our Circle Area Calculator.

Related Circle Measurements

When you know the radius and central angle, you can also find several other useful values:

• Sector Area (A) — the pie-shaped region enclosed by the arc and two radii. The formula is A = ½ × r² × θ.
• Chord Length (c) — the straight-line distance between the two endpoints of the arc. The formula is c = 2r × sin(θ/2). If you need to find the straight-line distance between any two points, our Distance Calculator can help.
• Diameter (d) — simply twice the radius: d = 2r.

Because all of these values are connected, knowing any two independent measurements lets you calculate the rest. For instance, if you know the arc length and the radius, you can work backward to find the central angle, and then find the chord length and sector area from there. Working with triangles formed by the chord and radii? Our Right Triangle Calculator and Triangle Area Calculator are useful companion tools.

Arc Length of Any Curve

Arc length is not limited to circles. You can find the length of any smooth curve described by a function y = f(x) over an interval from a to b. The formula uses calculus:

L = ∫_a^b √(1 + [f'(x)]²) dx

In plain words, you take the derivative of the function (which tells you the slope at every point), square it, add 1, take the square root, and then add up all those tiny values across the interval. This integral is often hard or impossible to solve by hand, so numerical methods like Simpson's Rule are used to get a very accurate approximation by breaking the curve into many small segments. If you need help computing derivatives separately, try our Derivative Calculator. Understanding the Slope Calculator can also be helpful, since the derivative represents the slope at each point along the curve. For problems involving how quickly a value changes over an interval, the Rate of Change Calculator is another handy resource.

Common Uses of Arc Length

Arc length comes up in many real-world situations. Engineers use it to calculate the length of curved roads, railway tracks, and bridges. In physics, arc length helps describe the path an object travels along a curved trajectory — for example, when analyzing projectile motion or the swing of a pendulum. Architects and designers use it when working with arches, domes, and circular structures. Even on a map, the distance between two cities along the curved surface of the Earth is an arc length calculation. In construction, understanding curved measurements is important when designing features like circular patios or curved walls, similar to how you'd use a Square Footage Calculator for flat areas.

Degrees vs. Radians

One common source of mistakes is mixing up degrees and radians. A full circle is 360° or 2π radians (about 6.2832 radians). The arc length formula L = r × θ only works when θ is in radians. If you use degrees by accident, your answer will be wrong. Always check your angle unit before calculating. Other angle units like gradians (where a full circle is 400 grad), arcminutes, and arcseconds also exist and can be converted to radians for use in the formula. If you're working with other geometric shapes and need related calculations, explore our Hexagon Calculator, Sphere Volume Calculator, Cylinder Volume Calculator, or Cone Volume Calculator for additional geometry tools. For finding the point exactly halfway along a chord or line segment, our Midpoint Calculator is a quick solution.

Frequently Asked Questions

What is arc length?

Arc length is the distance along the curved part of a circle (or any curve) between two points. Instead of measuring straight across, you measure along the curve itself, like laying a string on the curved path and then measuring the string.

What formula does this calculator use for arc length?

For circles, it uses L = r × θ, where r is the radius and θ is the central angle in radians. For general curves described by y = f(x), it uses L = ∫√(1 + [f'(x)]²) dx over your chosen interval.

How many values do I need to enter in the geometric tab?

You need to enter at least two independent values. For example, a radius and a central angle, or an arc length and a radius. The calculator will figure out all the other values from there. Note that radius and diameter count as one value since they are directly linked.

Can I use degrees instead of radians?

Yes. The calculator lets you pick your angle unit from the dropdown menu. You can choose degrees, radians, gradians, arcminutes, arcseconds, or milliradians. The tool converts your input to radians behind the scenes before calculating.

What units can I use for length and area?

For length, you can use millimeters, centimeters, meters, kilometers, inches, feet, yards, or miles. For sector area, you can use the squared versions of those same units (mm², cm², m², km², in², ft², yd², mi²). The calculator converts everything internally to meters for consistency.

What happens when I change the unit after entering a value?

The calculator automatically converts your number to the new unit. For example, if you entered 10 meters for the radius and then switch to feet, it will change the number to the equivalent value in feet. Your actual measurement stays the same.

How are radius and diameter connected in this tool?

They are auto-synced. When you type a radius, the diameter field updates to twice that value. When you type a diameter, the radius updates to half. They count as one known value, not two, so you still need to enter a second independent measurement.

What is sector area?

Sector area is the area of the pie-shaped slice of a circle formed by two radii and the arc between them. The formula is A = ½ × r² × θ, where θ is in radians. This calculator finds it automatically when you have enough information.

What is chord length?

Chord length is the straight-line distance between the two endpoints of an arc. It cuts directly across the circle rather than following the curve. The formula is c = 2r × sin(θ/2).

What does the Parametric / Function tab do?

It calculates the arc length of any curve you define as y = f(x) over an interval [a, b]. You type in a math function, set the start and end points, and the calculator uses Simpson's Rule to estimate the arc length numerically.

What functions can I type in the parametric tab?

You can use basic operations (+, -, *, /, ^) and common math functions like sin(x), cos(x), tan(x), sqrt(x), log(x), ln(x), exp(x), and abs(x). You can also use pi and e as constants.

What is Simpson's Rule?

Simpson's Rule is a method for estimating the value of an integral (area under a curve). It breaks the interval into many small segments and uses curved approximations to get a very accurate result. The more segments you use, the more accurate the answer.

How many integration segments should I use?

The default of 1,000 works well for most functions. If your function changes rapidly or you need extra precision, you can increase it up to 100,000. Higher numbers give more accuracy but take slightly longer to compute.

Why does my parametric calculation say the function produces undefined values?

This means the function cannot be evaluated at some point in your interval. For example, sqrt(x) is undefined for negative x, and 1/x is undefined at x = 0. Adjust your interval so the function is defined at every point within it.

Can the arc length be longer than the chord length?

Yes, always. The arc follows the curve, which is a longer path than the straight line (chord) between the same two points. The only time they would be equal is for an angle of zero, which means no arc at all.

What is the arc length of a full circle?

The arc length of a full circle is the circumference, which equals 2πr. If you set the central angle to 360° (or 2π radians), the calculator will give you the full circumference.

Can I enter the arc length and sector area to find the radius and angle?

Yes. The calculator can work backward from many combinations of two known values. If you enter arc length and sector area, it will solve for the radius and central angle, then fill in everything else.

Why does my angle need to be positive and at most 360°?

An arc on a circle is defined by a central angle between 0° and 360° (0 to 2π radians). An angle of 0 means no arc, and 360° means the full circle. Negative or larger angles don't make sense for a single arc on a circle.