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Arc Length and Sector Area Calculator

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.

Arc length, sector area, chord and sector perimeter from a radius and an angle, with the fraction of the full circle shown as its own step.

Formula and sources checked · How we check

Radius 10 ft, Angle 60 °, What you need Arc length — the curved edge

10.472

Result for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.

It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.

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

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Worked example

A 60° sector of a 10 ft circle has a 10.472 ft arc and covers 52.36 sq ft. Sixty degrees is a sixth of the circle, so both figures are exactly a sixth of the full 62.83 ft circumference and 314.16 sq ft area — the fraction is the whole method.

How to work it out yourself

  1. 1.Measure or calculate the radius — half the diameter.
  2. 2.Divide the angle by 360 to get the fraction of the circle the sector covers.
  3. 3.Multiply the full circumference by that fraction for the arc, or the full area for the sector.
  4. 4.The chord is the straight line across the open end, which is not the same as the arc.

The formula

  1. Fraction of the full circle60 / 360
  2. Full circumference2 * pi * 10
  3. Arc length62.831853 * 0.1666667
  4. Full circle areapi * pow(10, 2)
  5. Sector area314.15927 * 0.1666667
  6. Chord across the sector2 * 10 * sin(60 * pi / 360)
  7. Sector perimeter10.471976 + 2 * 10
  8. Answer10.471976 * 1 + 52.359878 * 0

Source: Wolfram MathWorld — circular sector, NIST Digital Library of Mathematical Functions

Questions people actually ask

What is the difference between the arc and the chord?
The arc is the curved edge and the chord is the straight line between its ends. For a 60° sector of a 10 ft circle the arc is 10.472 ft and the chord 10 ft — the chord is always shorter.
How do I work in radians instead?
Multiply the radius by the angle in radians for the arc, and half the radius squared by the angle for the area. Degrees divided by 360 does the same job without converting.

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