Arc Length Calculator

Arc Length and Sector Area Formulas

Arc length represents the distance along a circular arc. The relationship between arc length, radius, and central angle forms the foundation of these calculations. Similarly, sector area describes the region enclosed by two radii and the connecting arc.

L = r × θ

A = r² × θ ÷ 2

c = 2 × r × sin(θ ÷ 2)

L — Arc length
r — Radius of the circle
θ — Central angle in radians
A — Area of the sector
c — Chord length (straight line connecting arc endpoints)

Understanding the Arc Length Relationship

For a complete circle, the central angle equals 2π radians and the arc length equals the circumference (2πr). Any arc is simply a proportional fraction of this full circle. When the central angle θ (in radians) is multiplied by the radius r, you get the arc length. This elegant relationship holds because radians are dimensionless units derived from the ratio of arc length to radius.

If your angle is given in degrees, convert to radians first by multiplying by π/180. Alternatively, use the relationship that 360° corresponds to 2π radians.

Radians simplify calculations: Working in radians avoids conversion factors and makes the formula direct multiplication.
Sector area uses the same angle: The factor of one-half appears because a sector is half of the rectangle formed by two radii and an arc.
Chord length differs from arc length: The chord is a straight line between endpoints, always shorter than the curved arc (except at 0° where they coincide).

Recovering Missing Values

You don't always have both radius and angle. The calculator solves for missing inputs using algebraic rearrangement.

Finding radius from sector area and angle: Rearrange A = r² × θ ÷ 2 to get r = √(2A ÷ θ).

Finding angle from arc length and radius: Divide arc length by radius: θ = L ÷ r (result in radians).

Using chord length: If you know the chord and radius, the relationship c = 2r × sin(θ÷2) lets you solve for θ using inverse sine, then find arc length. This is useful in surveying and construction when direct angle measurement is difficult.

Using sector area and arc length: From A = r × L ÷ 2, you can derive r = 2A ÷ L, then calculate θ = L ÷ r.

Practical Applications

Arc length calculations appear across multiple fields. In engineering, designers calculate material length needed for curved ducts, pipes, or cable runs. Architects use sector areas to estimate paving or roofing material for curved surfaces. Mechanical engineers analyzing rotating machinery need arc length to determine distances traveled by points on rotating shafts.

Surveying uses these formulas to measure land parcels with curved boundaries. In astronomy, arc length describes angular distances across the sky. Road design applies these principles to calculate the length of curves and the area of traffic circles.

Common Pitfalls and Considerations

Avoid these frequent mistakes when working with arcs and sectors.

Angle units matter — The formulas L = r × θ and A = r² × θ ÷ 2 require θ in radians. If you're given degrees, multiply by π/180 first. Many calculation errors stem from mixing radians and degrees without conversion.
Arc length vs. chord confusion — The arc follows the curved path; the chord cuts straight across. For small angles they're nearly equal, but at 90° a radius-1 arc is π/2 ≈ 1.57 while the chord is √2 ≈ 1.41. Always clarify which distance the problem requires.
Sector area ≠ triangle area — A sector includes the curved arc region, not just the triangle formed by two radii. The sector area formula A = r² × θ ÷ 2 is always less than or equal to the full circle area πr², unlike triangles which can exceed sector bounds.
Unit consistency — If radius is in centimetres, arc length results in centimetres and sector area in square centimetres. Converting between metric and imperial mid-calculation introduces errors. Establish units upfront and maintain them throughout.

Frequently Asked Questions

Can I calculate arc length if I only know the chord and radius?

Yes. Use the relationship chord = 2r × sin(θ÷2), where θ is the central angle in radians. Divide the chord by 2r, then take the inverse sine to find θ÷2. Double that result to get θ in radians. Multiply θ by the radius to find arc length. This method is valuable in surveying when angles are hard to measure directly but the straight-line chord distance is known.

Does arc length always require radians?

Arc length itself is a distance measured in metres, centimetres, feet, or any length unit—never in radians. However, the formula L = r × θ demands the central angle θ in radians. You can express the original angle in degrees, but convert to radians before calculation. Once you have the arc length result, it's purely a length, without angular units.

How do I find the central angle if I know arc length and radius?

Divide the arc length by the radius: θ = L ÷ r. This gives the angle in radians. To convert to degrees, multiply by 180/π. This is the inverse of the standard arc length formula and is useful in reverse-engineering problems where the arc distance is measured physically but the angle is unknown.

What's the difference between arc length and sector area?

Arc length is a one-dimensional measurement—the distance along the curved path. Sector area is two-dimensional—the region enclosed by two radii and the arc. A large radius with a small angle might have significant arc length but small area. Conversely, a large angle with a small radius can yield small arc length but measurable area. Both use the same central angle but answer different geometric questions.

If I double the radius, how does the arc length change?

Arc length is directly proportional to radius. Doubling the radius doubles the arc length, assuming the central angle stays constant. This is why the formula is simple multiplication: L = r × θ. In contrast, sector area scales with the square of the radius, so doubling the radius quadruples the area.