Arc Length Formula Calculator: Find a Circle’s Arc

An arc length formula calculator finds the distance along part of a circle when you enter its radius and central angle. Enter the radius in any length unit and the angle in degrees; the calculator returns the arc length in that same length unit.

Formula

For an angle measured in degrees, arc length is the circle’s circumference multiplied by the fraction of a full turn covered by the angle:

s = (a / 360) × 2πr = πra / 180

• s is the arc length, in the same unit used for the radius.
• r is the circle’s radius, in units such as centimetres, metres, inches, or feet.
• a is the central angle, in degrees.
• π is approximately 3.14159.

The central angle is the angle formed at the circle’s centre by lines drawn to the two endpoints of the arc. The calculator’s second result, the full circumference, is 2πr; it provides a useful check on the arc length.

Worked Example

Using the calculator’s default inputs, the radius is 5 units and the central angle is 60 degrees. The angle covers 60/360, or one-sixth, of a full circle.

1. Find the full circumference: 2π × 5 = 10π ≈ 31.4159 units.
2. Take one-sixth of that circumference: 31.4159 × (60/360) ≈ 5.2360 units.
3. Round the arc length to two decimal places: 5.24 units.

The circumference output is 31.42 units when rounded to two decimal places. The arc is shorter than the full circumference, as it should be for a 60-degree angle.

How to Use the Calculator

Measure or identify the circle’s radius, then enter the angle at its centre between the arc’s endpoints. The output uses the same length unit as the radius: a radius of 5 centimetres gives an answer in centimetres, while a radius of 5 feet gives an answer in feet. The word “units” in the calculator is a placeholder, not a required measurement system.

• If you are given the diameter, divide it by 2 to get the radius.
• If your angle is in radians, convert it to degrees before using this calculator: degrees = radians × 180/π.
• For a reflex arc, whose central angle is greater than 180° and less than 360°, enter that larger angle if you want the longer arc.
• Use 360° for one complete revolution; the arc length should then equal the circumference.

The calculator accepts nonnegative radius and angle values. A radius of zero gives an arc length of zero; for an ordinary circle, use a positive radius. An angle of zero also gives a zero-length arc.

Degrees, Radians, and the Alternative Formula

The degree-based formula is convenient when angles are given in degrees, but many mathematics and science problems use radians. In radians, the formula is simpler:

s = rθ

Here, θ is the central angle in radians, and r and s use the same length unit. For example, a 60-degree angle is π/3 radians. With a radius of 5 units, s = 5 × π/3 ≈ 5.24 units, matching the degree-based calculation.

Radians are defined by the relationship between arc length and radius: θ = s/r. One radian is the angle that subtends an arc equal in length to the radius. A full circle is 2π radians, equivalent to 360 degrees. Do not put a degree value directly into s = rθ; convert it first or use the degree formula.

When Arc Length Is Useful

Arc length describes distance along a curved path, not the straight-line distance between its endpoints. It appears in geometry, construction, machining, navigation, and circular-motion problems. For instance, it can help determine how much material is needed to follow part of a circular edge, or how far a point on a rotating wheel travels.

For motion around a circle, the arc length is the distance travelled along the circular path. If you also know the time taken, average tangential speed is arc length divided by elapsed time. The angle and radius alone do not determine speed; time is needed for that calculation.

Arc length is also different from the area of a sector. A sector is the wedge-shaped region enclosed by two radii and an arc. For an angle in degrees, its area is (a/360) × πr², measured in square units. Arc length is a one-dimensional distance, measured in units such as metres or inches.

Common Mistakes and Limits

• Using diameter as radius: Substituting the diameter doubles the calculated arc length. Halve the diameter first.
• Mixing units: Keep the radius in one consistent length unit. If a radius is provided in inches but the result is needed in feet, convert the input or convert the output; 12 inches equal 1 foot.
• Confusing degrees and radians: The degree formula uses a/360, while s = rθ requires θ in radians.
• Choosing the wrong arc: Two endpoints divide a circle into a shorter minor arc and a longer major arc. Their central angles add to 360°. Make sure the entered angle corresponds to the arc you need.
• Rounding too early: Keep extra digits during intermediate calculations, then round the final length to a precision that suits the measurement.

The formula assumes a circular arc with a fixed radius. It does not directly calculate the length of a general curved path whose curvature changes. In practical measuring or manufacturing, imperfections and measurement uncertainty can also make the actual length differ slightly from the ideal mathematical result.

Frequently Asked Questions

How do you calculate arc length?
For an angle in degrees, use s = πra/180, where r is the radius and a is the central angle. For an angle in radians, use s = rθ.

Can I find arc length from diameter and angle?
Yes. First divide the diameter by 2 to find the radius, then apply the arc-length formula. With a degree angle a and diameter d, the equivalent expression is s = πda/360.

Does arc length use the same units as radius?
Yes. If the radius is entered in centimetres, the arc length is in centimetres; if it is in metres, the result is in metres. The angle does not have a length unit.

What is the arc length of a full circle?
A full circle has a central angle of 360 degrees, or 2π radians. Its arc length is the circumference, 2πr.

Is arc length the straight distance between two points?
No. Arc length follows the curved edge of the circle. The straight distance between the endpoints is a chord, which is generally shorter than the corresponding arc.

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in r | Radius | units | 5 | 0
in a | Central angle | degrees | 60 | 0
out s | Arc length | units | pi * r * a / 180 | 2
out c | Full circumference | units | 2 * pi * r | 2