Circular Sector Area
A = (θ/360)·π·r²P = 2r + (θ/360)·2π·rarc = (θ/360)·2π·rIt is the slice of a circle bounded by two radii and the arc between them — the shape of a pizza slice.
The area is the fraction of the circle matching the angle: the angle divided by 360 degrees, multiplied by the circle’s total area.
The perimeter adds both radii to the arc length. That is the part that usually confuses: the outline of the slice includes the two straight cuts, not just the curved edge.
A sector is the whole slice, with its vertex at the centre. A circular segment is only the part between the chord and the arc — the slice minus the central triangle.
If you need the area of a partly filled gutter, or the glass above a straight line in a round window, what you want is the segment, not the sector.
Confusing the two is the most common cause of a wrong result in this calculation.
- Pie chart slices, where each sector represents a proportion.
- The area swept by a swinging gate or a door, to check clearance.
- Sprinkler coverage with a sectored jet, watering only part of the circle.
- Mechanical parts such as partial gears, cams and pulleys with an angular cut.
Frequently asked questions
The sector is the complete slice with its vertex at the centre. The segment is only the part between the chord and the arc, without the central triangle. For a partly filled gutter, use the segment.
Yes. The outline of the slice is the two straight cuts plus the arc. Using only the arc length badly underestimates the perimeter.
It is the same fraction applied to the circumference: the angle divided by 360, times two π times the radius.
It works normally. The sector becomes the larger part of the circle, and the formula stays valid up to 360 degrees, where the result is the whole circle.