# Arc Length Calculator

Calculate arc length, sector area, and chord length from radius and central angle. Supports degrees and radians. Free online arc length calculator.

## What this calculates

Calculate the arc length, sector area, and chord length of any circular arc. Enter the radius and central angle (in degrees or radians) for instant results.

## Inputs

- **Radius** — The radius of the circle.
- **Angle Unit** — options: Degrees, Radians — Whether the angle is in degrees or radians.
- **Central Angle** — The central angle that subtends the arc.

## Outputs

- **Arc Length** — The length of the arc.
- **Sector Area** — The area of the circular sector.
- **Chord Length** — The straight-line distance between the arc endpoints.
- **Angle (degrees)** — The central angle in degrees.
- **Angle (radians)** — The central angle in radians.

## Details

An arc is a portion of the circumference of a circle, and the arc length depends on the radius and the central angle that subtends it.

Arc Length Formulas

- Arc length: s = r x theta (theta in radians)

- Arc length: s = (theta/360) x 2 x pi x r (theta in degrees)

- Sector area: A = (1/2) x r^2 x theta (theta in radians)

- Chord length: c = 2 x r x sin(theta/2)

Degrees vs. Radians

Radians are the natural unit for angle measurement in mathematics. One full revolution = 2pi radians = 360 degrees. The simplicity of arc length in radians (s = r x theta) is one reason radians are preferred in calculus and physics.

Arc length calculations are used in engineering (conveyor belts, gear teeth), construction (curved walls, arched bridges), and navigation (great circle distances on Earth).

## Frequently Asked Questions

**Q: How do I calculate arc length?**

A: If the angle is in radians: arc length = radius x angle. If in degrees: arc length = (angle/360) x 2 x pi x radius. For example, a circle with radius 10 and central angle 45 degrees: s = (45/360) x 2 x pi x 10 = (1/8) x 62.83 = 7.854 units.

**Q: What is a sector of a circle?**

A: A sector is the pie-shaped region enclosed by two radii and an arc. Its area is a fraction of the full circle's area, proportional to the central angle: A = (theta/360) x pi x r^2 in degrees, or A = (1/2) x r^2 x theta in radians. A semicircle is a sector with a 180-degree angle.

**Q: What is the difference between arc length and chord length?**

A: Arc length is the distance along the curved arc of the circle, while chord length is the straight-line distance between the two endpoints of the arc. The arc is always longer than the chord (unless the angle is 0). As the angle increases toward 180 degrees, the chord approaches the diameter while the arc approaches half the circumference.

**Q: How do I convert between degrees and radians?**

A: To convert degrees to radians: multiply by pi/180. To convert radians to degrees: multiply by 180/pi. Key values: 360 degrees = 2pi radians, 180 degrees = pi radians, 90 degrees = pi/2 radians, 60 degrees = pi/3 radians, 45 degrees = pi/4 radians.

**Q: Why are radians used instead of degrees in calculus?**

A: Radians simplify many formulas. The derivative of sin(x) is cos(x) only when x is in radians. The arc length formula s = r x theta is simplest in radians. The Taylor series for trigonometric functions require radian inputs. Radians are a dimensionless ratio (arc length divided by radius), making them the natural choice.

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Source: https://vastcalc.com/calculators/math/arc-length
Category: Math
Last updated: 2026-04-21