Circle Sector Calculator
Calculate arc length, chord length, sector area, segment area and the sagitta of a circle from any two known measurements, with degrees and radians throughout.
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A sector and a segment are not the same
The sector is the pie slice bounded by two radii and the arc. The segment is the smaller region cut off by the chord alone, between the chord and the arc. The segment area is the sector area minus the triangle formed by the two radii and the chord, and the two differ substantially at small angles: a ten degree sector is nearly a thin triangle while its segment is a sliver a fraction of the size. Using one where the other is meant is the most common error in this calculation.
The sagitta is what most practical problems need
The sagitta, or rise, is the perpendicular distance from the middle of the chord to the arc. It is what a machinist measures to find the radius of an existing curve, what determines the depth of a curved cut, and what sets the height of an arch from its span. Combined with the chord length it determines the radius completely, and that combination is easier to measure on a real object than the radius itself, which requires finding the centre.
Radians are not an inconvenience
Arc length is simply the radius times the angle in radians, and sector area is half the radius squared times the same angle. In degrees both formulas carry a conversion factor that has to be remembered and is easy to misplace. This is the whole reason radians exist as a unit: they make the relationships between angle, arc and area direct rather than scaled. Both units are shown here, with the radian forms being the ones worth memorising.
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Frequently Asked Questions
What is the difference between a sector and a segment?
A sector is the pie slice between two radii and the arc. A segment is the smaller region between a chord and the arc. The segment is the sector minus the triangle formed by the radii and the chord, and at small angles it is a small fraction of the sector.
What is the sagitta?
The perpendicular distance from the midpoint of the chord to the arc, also called the rise. It is what a machinist measures to determine the radius of an existing curve, since the chord and the sagitta together fix the radius without needing to find the centre.
Why are the formulas simpler in radians?
Because arc length is exactly the radius times the angle and sector area is exactly half the radius squared times the angle, with no conversion factor. That directness is the reason the unit exists, and the degree versions carry a factor that is easy to misplace.
Can I find the radius from a chord and a sagitta?
Yes. The radius equals the sagitta halved plus the chord squared divided by eight times the sagitta. This is the standard workshop method for measuring the radius of an existing arc, since it needs only a straight edge and a depth gauge.
What happens beyond 180 degrees?
The sector becomes the larger portion of the circle and the segment becomes the major segment, larger than a semicircle. The formulas continue to hold, and the chord length starts decreasing again past 180 degrees, since the two endpoints come back together.
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How to Use
Enter a radius and an angle, or any two measurements, to get every sector and segment value.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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