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Arc Length Calculator

Length of the curved edge of a sector, from the radius and the angle, with the fraction of the full circle shown as a step of its own rather than folded in.

The arc is the curved edge, not the straight line across the open end. For a 60° sector of a 10 ft circle the arc is 10.472 ft and that chord is 10 ft — close enough to confuse and far enough to cut material short.

With radius 10 ft, angle 60 °, arc length and sector area comes to 10.472 — result. It is reached in 8 steps, the last of which is 10.471976 * 1 + 52.359878 * 0, and each one is printed on the page with its numbers filled in. The formula is the one published by Wolfram MathWorld, not an approximation fitted to it.

Formula and sources checked · How we check

Result
10.472
Fraction of the full circle
60 / 3600.167
Full circumference
2 * pi * 1062.832 ft
Arc length
62.831853 * 0.166666710.472 ft
Full circle area
pi * pow(10, 2)314.159 sq ft
Sector area
314.15927 * 0.166666752.36 sq ft
Chord across the sector
2 * 10 * sin(60 * pi / 360)10 ft
Sector perimeter
10.471976 + 2 * 1030.472 ft
Answer
10.471976 * 1 + 52.359878 * 010.472

What if you know the arc and want the angle?

Reverse it: the angle is the arc divided by the radius, which gives radians, times 180/π for degrees. A 12 ft arc on a 10 ft radius is 1.2 radians, so 68.75°. That is the useful direction when the curve exists on site and the drawing does not.

Does the formula hold for a whole circle?

At 360° the arc is the entire circumference and the formula returns exactly 2πr. That is the sense check worth running on any arc figure: it should be the same fraction of 2πr as the angle is of 360, and if it is not, the angle went in as radians.

The formula

  1. Fraction of the full circleangle / 360
  2. Full circumference2 * pi * radius
  3. Arc lengthcircumference * fraction
  4. Full circle areapi * pow(radius, 2)
  5. Sector areafull_area * fraction
  6. Chord across the sector2 * radius * sin(angle * pi / 360)
  7. Sector perimeterarc + 2 * radius
  8. Answerarc * want_arc + area * want_area

Formula from Wolfram MathWorld — circular sector, NIST Digital Library of Mathematical Functions.

The same calculator, other cases

Sector Area Calculator

Undecided? The Arc Length and Sector Area Calculator carries every case behind one control.

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