Arc Length and Sector Area Calculator
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.
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 arc * want_arc + area * want_area, 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
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.
- Fraction of the full circle
angle / 3600.167- Full circumference
2 * pi * radius62.832 ft- Arc length
circumference * fraction10.472 ft- Full circle area
pi * pow(radius, 2)314.159 sq ft- Sector area
full_area * fraction52.36 sq ft- Chord across the sector
2 * radius * sin(angle * pi / 360)10 ft- Sector perimeter
arc + 2 * radius30.472 ft- Answer
arc * want_arc + area * want_area10.472
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.Measure or calculate the radius — half the diameter.
- 2.Divide the angle by 360 to get the fraction of the circle the sector covers.
- 3.Multiply the full circumference by that fraction for the arc, or the full area for the sector.
- 4.The chord is the straight line across the open end, which is not the same as the arc.
The formula
- Fraction of the full circle
angle / 360 - Full circumference
2 * pi * radius - Arc length
circumference * fraction - Full circle area
pi * pow(radius, 2) - Sector area
full_area * fraction - Chord across the sector
2 * radius * sin(angle * pi / 360) - Sector perimeter
arc + 2 * radius - Answer
arc * want_arc + area * want_area
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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<iframe src="https://rulecalculators.com/embed/circle-sector" width="100%" height="420" style="border:1px solid #e7e4de;border-radius:12px" title="Arc Length and Sector Area Calculator"></iframe>Straight to your case
The control above covers every case. These are the same calculator fixed to one of them, with the formula for that case written out.
- Arc Length CalculatorLength of the curved edge of a sector, from the radius and the angle, with the fraction of the full circle shown as its own step.
- Sector Area CalculatorArea of a wedge cut from a circle, from the radius and the angle — the same fraction applied to the area rather than to the circumference.