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Sector Area Calculator

Area 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, with each step shown.

A sector is a fraction of the circle, and the fraction is the whole method: 90° is a quarter, so the area is a quarter of the full circle. Nothing about the formula changes with the size of the circle.

With radius 10 ft, angle 60 °, arc length and sector area comes to 52.360 — result. It is reached in 8 steps, the last of which is 10.471976 * 0 + 52.359878 * 1, 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
52.360
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 * 0 + 52.359878 * 152.36

What is the difference between a sector and a segment?

A sector is the wedge cut in to the centre, with both straight edges included. A segment is what a straight chord slices off and has no point at the centre. Subtract the triangle from the sector to get it: at 60° on a 10 ft circle, 52.36 less 43.30 leaves 9.06 sq ft.

How exact does the angle have to be?

As exact as you need the area to be, because area scales straight with angle: a 5° error on a 60° sector is 8% of the answer. The radius is less forgiving, since it is squared — a 5% error there is 10% of the area, and it compounds with the angle rather than cancelling it.

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

Arc Length Calculator

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

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