Arc Length Calculator
Calculate the exact length of any circular arc by entering the radius and central angle in degrees. Useful for engineering, geometry, and design tasks involving curved paths or sectors.
About this calculator
The arc length is the distance along the curved edge of a circle between two points. It depends on two quantities: the radius of the circle and the central angle that subtends the arc. Because the full circumference of a circle is 2πr (covering 360°), a fraction of that circle corresponds to a proportionally shorter arc. The formula is: Arc Length = r × θ, where r is the radius and θ is the central angle expressed in radians. To convert degrees to radians, multiply by π/180, giving the full formula: Arc Length = r × (central_angle × π / 180). This formula is used in civil engineering for road curves, in manufacturing for cutting curved materials, and in everyday geometry problems involving circles.
How to use
Suppose a circular garden path has a radius of 5 meters and sweeps through a central angle of 60°. Step 1: Convert the angle to radians: 60 × π / 180 = 1.0472 radians. Step 2: Multiply by the radius: Arc Length = 5 × 1.0472 = 5.236 meters. So the curved path is approximately 5.24 meters long. Enter radius = 5 and central angle = 60 into the calculator to confirm this result instantly.
Frequently asked questions
What is the difference between arc length and chord length?
Arc length is the distance measured along the curved edge of the circle between two points, while chord length is the straight-line distance connecting those same two points. Arc length is always greater than or equal to chord length, with equality only when the angle is 0°. For large central angles, the difference between the two becomes significant. Use arc length when you need the actual path distance along the curve.
How do I calculate arc length when the angle is given in radians?
If your central angle is already in radians, simply multiply it directly by the radius: Arc Length = r × θ. There is no need to apply the π/180 conversion factor, since that step only applies when the angle is in degrees. For example, an angle of π/3 radians with a radius of 4 gives Arc Length = 4 × (π/3) ≈ 4.19 units. This calculator accepts degrees and handles the conversion automatically.
When would I use an arc length calculator in real life?
Arc length calculations appear in many practical fields. Civil and road engineers use them to determine the length of curved road segments or railway tracks. Architects and designers use arc lengths when planning circular staircases, arched windows, or curved facades. Machinists and fabricators need arc length to cut the correct amount of curved material. Even GPS and navigation systems use arc-length concepts to measure distances along curved paths on Earth's surface.