How to find arc length
Arc length is the portion of a circle's circumference swept by a central angle: arc = 2π × r × (angle ÷ 360) for an angle in degrees. A 90° arc (a quarter circle) of radius 5 is 2π × 5 × 0.25 = about 7.85 units.
The formula is easy to remember once you see where it comes from. A full circumference is 2πr, and a 360° angle sweeps all of it, so any other angle sweeps exactly the fraction angle ÷ 360 of it. The angle has to be in radians for the formula to work, so convert it first with the degrees to radians calculator.
How to use this calculator
Enter the radius and the central angle in degrees. The result is the arc length in the same unit as the radius. A full 360° angle returns the entire circumference.
Two mistakes cause most wrong answers: entering a diameter where a radius is expected, which doubles the result, and typing a radian value into the degrees field, which shrinks it about 57-fold. If you want the wedge of area the arc encloses rather than the curve itself, use the sector area calculator.
Radians instead of degrees
If your angle is in radians, the formula is simply arc = r × θ. To convert, degrees × π ÷ 180 = radians. The whole circle's perimeter (the circumference) comes from the circle diameter calculator.
The radian version is shorter for a reason: a radian is defined as the angle subtending an arc equal to the radius, so θ radians always subtend θ radii of arc, with no conversion factor at all.
Frequently asked questions
- What is the formula for arc length?
- Arc = 2π × r × (angle ÷ 360) for degrees, or arc = r × θ for an angle θ in radians.
- What is the arc length of a 90° arc with radius 5?
- About 7.85 units — a quarter of the circumference 2π × 5.
- How do I convert degrees to radians?
- Multiply degrees by π and divide by 180. So 90° = π ÷ 2 ≈ 1.5708 radians.
- What angle gives the full circumference?
- 360 degrees (or 2π radians) returns the entire circumference of the circle.
- What units does the arc length come out in?
- The arc length is in exactly the same unit as the radius you entered, because the angle fraction is a pure number with no units. A radius in centimetres gives an arc in centimetres, and a radius in inches gives an arc in inches. Never mix units between the radius and the answer you report.