Arc Length Calculator - Find Arc Length & Sector Area
Find arc length, sector area, chord length, and angle conversions from a radius and central angle.
Arc Length Calculator
Results
What is an Arc Length Calculator?
An Arc Length Calculator helps you find the distance along a curved section of a circle. Instead of measuring the straight-line width across the shape, this tool measures the actual path of the arc, which is the part of the circumference between two points.
That makes it useful in geometry classes, design work, construction layouts, and any situation where a curved edge matters. If you are describing a track, a circular window, a pipe bend, or a pie-slice shape, arc length is usually the number you need before you can size materials correctly.
The same measurement shows up in practical jobs like planning a curved retaining wall, checking how much trim is needed around an arch, or estimating the length of a decorative border around a circular feature. In those cases, the straight-line chord is not enough because the material follows the curve, not the shortcut.
This calculator is especially helpful for:
- Students who need quick, reliable answers for circle geometry problems.
- Designers who work with curved edges, arcs, and circular layouts.
- Builders and makers who need curved measurements before cutting material.
- Anyone converting angle units between degrees and radians for circle formulas.
Arc length is easy to misread when a problem mixes radius, diameter, degrees, and radians in the same sentence. This tool keeps the inputs focused on the two values that matter most, then translates them into the measurements you can actually use.
If you need the full circle instead of a curved segment, try our Circle Calculator.
How the Arc Length Calculator Works
The calculation starts with the radius and the central angle of the arc. The core formula is simple: s = rθ, where s is arc length, r is radius, and θ is the angle in radians. When the angle is entered in degrees, the calculator converts it automatically.
Once the angle is in radians, the formulas stay direct and proportional to the radius.
OpenStax's arc-length section shows the same rule in practice: the angle must be in radians before you use the circle formulas.
According to OpenStax Precalculus 2e, 5.1 Angles, the arc-length and sector-area formulas are evaluated from radian measure, not raw degree input.
Once the angle has been converted to radians, the calculator also derives the sector area, the chord length, and the fraction of the full circle. That is useful because circle problems often ask for several related values, not just a single answer.
Example: if the radius is 10 cm and the angle is 60 degrees, the calculator first converts the angle to 1.0472 radians. It then returns an arc length of about 10.472 cm, a sector area of about 52.36 cm², and a chord length of about 10 cm. That makes the formulas easier to understand in a real scenario.
The same logic scales cleanly to larger inputs. A 120-degree arc on a 25-foot radius creates a much longer curve, but the process is identical: convert the angle if needed, multiply by the radius, and use the same angle to calculate the sector area.
If you want to move between angle units before solving a geometry problem, use our Triangle Calculator.
Key Concepts Explained
Radius
The radius is the straight-line distance from the center of the circle to the edge. It is the starting point for both arc length and sector area calculations.
Central Angle
The central angle is the angle formed at the center of the circle by the two radii. A larger angle means a longer arc and a larger sector.
Radians
Radians connect angle and radius directly. In radians, the angle measure is based on the circle itself, which makes the formulas shorter and easier to use.
Sector Area
The sector is the pie-slice section bounded by two radii and the arc. Its area helps you estimate enclosed space, not just curved distance.
Chord Length
The chord is the straight line connecting the arc’s endpoints. It is always shorter than the arc unless the angle is zero.
If you want to compare the arc with the full circle's area, use our Area Calculator to check the shape's overall footprint.
How to Use This Arc Length Calculator
Enter the radius
Type the radius of your circle in the first field. This can be any non-negative number in your chosen length unit.
Add the central angle
Enter the angle that cuts out the arc. Use degrees for everyday geometry or radians for direct formula work.
Choose the units
Select the unit that matches your measurements so the arc length and chord length are reported in the same system.
Review the related outputs
Use the results panel to check sector area, radians, chord length, and the fraction of the circle covered by the angle.
If the angle is a neat fraction of the circle, the interpretation is even easier. A 90-degree arc is one quarter of a circle, a 180-degree arc is a semicircle, and a 360-degree angle gives you the full circumference.
If you need to compare your arc against the full circle, our Length Converter can help you keep the radius and arc length in the same unit system.
Benefits of Using This Calculator
- • Fast answers: You do not need to convert degrees by hand or work through each formula separately.
- • Fewer mistakes: The calculator handles the degree-to-radian conversion automatically, which reduces common geometry errors.
- • Multiple outputs: You get arc length, sector area, chord length, and the fraction of the circle from one input set.
- • Clear context: The results show how much of the circle your angle covers, which makes the number easier to interpret.
- • Better planning: Curved layouts are easier to estimate when you know both the curved distance and the enclosed area.
- • Safer ordering: Knowing the curve length helps you buy the right amount of pipe, trim, edging, or flexible material the first time.
- • Cleaner study notes: The calculator gives you the radians conversion, which makes homework and formula checking much simpler.
For other geometry checks, the Polygon Area Calculator can help you compare curved and straight-sided shapes.
Factors That Affect Your Results
Angle size
Larger angles create longer arcs and larger sectors. A full 360-degree sweep gives the entire circumference.
Radius length
Arc length scales directly with radius, while sector area scales with the square of the radius, so bigger circles grow quickly.
Angle units
Using degrees or radians changes how the input is interpreted. The calculator converts degrees automatically, but the result is still based on radians.
Measurement precision
Small rounding differences in the radius or angle can produce noticeable changes, especially for large circles or wide arcs.
Reference angle
An angle measured from the center of the circle must match the arc you want. If you choose the wrong arc, the result can describe a different part of the circle entirely.
Unit consistency
The radius and chord length must use the same unit system. Mixing centimeters and inches will produce a number that looks correct but is physically wrong.
As a quick sanity check, the arc length should always be shorter than the full circumference unless you are using a 360-degree angle. If your result is larger than the circumference, the angle or units likely need another look.
When you want to express the arc as a fraction of a full circle, the Fraction Calculator can help you verify the ratio behind the curve.
OpenStax also explains that arc length is a fraction of the full circumference and sector area is a fraction of the circle. That matches the way this calculator turns one radius-angle pair into multiple related answers.
According to OpenStax Algebra and Trigonometry 7.1 Angles, the radian measure ties an angle directly to the circle, which is why degree input must be converted first. That is why s = rθ and A = 1/2 θr² are the core circle formulas used for these results.
Frequently Asked Questions (FAQ)
How do you calculate arc length?
Arc length is calculated with the formula s = rθ, where r is the radius and θ is the central angle in radians. If your angle is in degrees, convert it first by multiplying the degree measure by π/180.
What is the formula for arc length in degrees?
When the angle is given in degrees, use s = 2πr × (θ/360). That is the same circle formula rewritten for degree input, so the calculator can handle common classroom and real-world measurements without manual conversion.
How is sector area related to arc length?
Sector area and arc length come from the same central angle. Once you know the radius and angle, the sector area is A = 1/2 r²θ and the arc length is s = rθ, both using radians for θ.
Why do you convert degrees to radians?
Radians make the circle formulas direct and consistent. In radians, the angle is measured by the radius of the circle itself, so arc length and sector area use shorter formulas and avoid extra conversion mistakes.
Can arc length be longer than the radius?
Yes. Arc length can easily be longer than the radius because it measures a curved distance along part of the circle. A wide central angle or a large circle can produce an arc that is much longer than the radius.
What is the difference between arc length and chord length?
Arc length follows the curve of the circle, while chord length is the straight line between the same two points. The chord is always shorter unless the arc is essentially flat, which is why the two values are used for different planning tasks.