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 calculator finds the arc length. This calculator gives you an accurate answer for a geometry homework problem, a design project, or just understanding circles better.
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.
Related Circle Measurements
When you know the radius and central angle, you can also find several other useful values:
- Sector Area (A) is the pie-shaped region enclosed by the arc and two radii. The formula is A = ½ × r² × θ.
- Chord Length (c) is the straight-line distance between the two endpoints of the arc. The formula is c = 2r × sin(θ/2).
- Diameter (d) is 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.
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 = ∫ab √(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. The derivative represents the slope at each point along the curve.
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.
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.