MyTrigCalculator.com

Arc Length Calculator

Compute arc length, central angle, radius, sector area, and chord length for any circular arc — enter any two values.

Enter Any 2 Values

Formula

s = r × θ

(angle in radians)

Results

Enter at least 2 values to compute the arc

Arc Length Explained

The arc length is the distance along the curved part of a circle between two points. It's a fundamental quantity in trigonometry, calculus, and engineering — appearing whenever you measure something along a circular path: a road bend, a clock hand's tip, a satellite orbit segment, or the rim of a wheel.

The core formula

s = r × θ

(s = arc length, r = radius, θ = central angle in radians)

If your angle is in degrees, convert first: θ_rad = θ_deg × π / 180. So in degrees, the formula is s = r × θ_deg × π / 180.

Worked example

A circular running track has a radius of 50 meters. How long is a 90° arc?

θ_rad = 90 × π/180 = π/2 ≈ 1.5708
s = 50 × π/2 ≈ 78.54 meters

Related quantities

  • Chord length — straight-line distance between the arc's endpoints: chord = 2r × sin(θ/2)
  • Sector area — pie-slice area: A = ½ × r² × θ (radians)
  • Segment area — area between the chord and the arc: A = ½r²(θ − sin θ)

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Frequently Asked Questions

What is the formula for arc length?

Arc length = radius × central angle (in radians), or s = r × θ. If your angle is in degrees, convert it first: θ_rad = θ_deg × π/180. So in degrees: s = r × θ × π/180.

What's the difference between arc length and chord length?

The arc length is the curved distance along the circle's edge between two points. The chord is the straight-line distance between those same two points. The chord is always shorter than the arc; they're equal only when the angle approaches zero.

How do I find the chord length from the arc?

Chord = 2 × r × sin(θ/2), where θ is the central angle in radians. The calculator returns this automatically. The chord is always shorter than the arc unless the angle is zero.

What is sector area and how is it different from arc length?

Arc length is the curved boundary of a sector — a 1-dimensional length. Sector area is the area enclosed by the two radii and the arc — a 2-dimensional region. Sector area = ½ × r² × θ (with θ in radians).

Why are radians easier than degrees for arc length?

Because the arc length formula in radians is just s = r × θ — no conversion factors. With degrees you have to multiply by π/180. Radians were literally defined so that arc length on a unit circle equals the angle measure.

Can I find the radius if I know the arc length and angle?

Yes. Rearrange s = r × θ to get r = s / θ (with θ in radians). This calculator handles all three conversions: enter any 2 of {radius, angle, arc length} and it computes the third plus the chord and sector area.

Learn More About Trigonometry

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