What this calculator does
A sector is a wedge of a circle, bounded by two radii and the arc between them. Its area and arc length are both simple fractions of the whole circle, in proportion to the angle it spans: a 90° sector is exactly a quarter of everything.
A segment is different, and the distinction matters. It is the region between a chord and the arc, so a segment is a sector with the triangular part removed. Confusing the two is the most common error in this topic.
The formula
Sector area and arc length scale directly with the angle as a fraction of 360°. The chord comes from the isosceles triangle formed by the two radii, using twice the sine of half the angle. Segment area subtracts that triangle from the sector, which is why the formula involves the sine term.
| Term | Meaning |
|---|---|
| Sector | The wedge bounded by two radii and an arc. |
| Arc | The curved portion of the circumference. |
| Chord | The straight line joining the two arc endpoints. |
| Segment | The region between the chord and the arc. |
| Central angle | The angle at the centre spanning the sector. |
The inputs explained
| Field | What to enter |
|---|---|
| Radius | The radius of the circle. |
| Central angle (°) | The central angle in degrees. 360 gives the whole circle. |
When to use it
Cutting a pie chart or a wedge
A percentage of a whole converts to degrees by multiplying by 3.6. A 25 per cent slice is 90° and takes exactly a quarter of the area.
Calculating a curved path
The arc length is the distance travelled around a bend of given radius: the calculation behind road curves, running tracks and conveyor layouts.
Fluid in a horizontal tank
The cross-section of liquid in a partly filled cylindrical tank is a circular segment. The segment area gives the fill fraction, which is why tank gauges are non-linear.
Cutting curved material
The chord length is the straight-line distance across a curve, which is what a saw cut or a sheet width needs to accommodate.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Sector measurements at a radius of 10
A fixed radius of 10 with the central angle varying from a narrow wedge to a full circle.
| Central angle | Sector area | Arc length | Chord length | Segment area |
|---|---|---|---|---|
| 30° | 26.180 | 5.236 | 5.176 | 1.180 |
| 60° | 52.360 | 10.472 | 10.000 | 9.059 |
| 90° | 78.540 | 15.708 | 14.142 | 28.540 |
| 120° | 104.720 | 20.944 | 17.321 | 61.418 |
| 180° | 157.080 | 31.416 | 20.000 | 157.080 |
| 360° | 314.159 | 62.832 | 2.4493e-15 | 314.159 |
How radius scales a 60° sector
A fixed 60° wedge at increasing radii.
| Radius | Sector area | Arc length | Chord length |
|---|---|---|---|
| 1 | 0.5236 | 1.047 | 1.0000 |
| 5 | 13.090 | 5.236 | 5.000 |
| 10 | 52.360 | 10.472 | 10.000 |
| 20 | 209.440 | 20.944 | 20.000 |
| 50 | 1,309.00 | 52.360 | 50.000 |
| 100 | 5,235.99 | 104.720 | 100.000 |
Questions
What is the difference between a sector and a segment?
A sector is bounded by two radii and the arc, like a slice of pie. A segment is bounded by a chord and the arc, like the slice with its pointed end cut off. The segment is always the smaller of the two.
How do I convert a percentage to a central angle?
Multiply by 3.6, since 100 per cent corresponds to 360°. A 35 per cent share is 126°.
How do I find arc length in radians?
Arc length equals radius times the angle in radians, which is the cleanest form of the relationship. In degrees the conversion factor of π ÷ 180 has to be carried through.
Why does the segment formula involve a sine?
Because the segment is the sector minus the triangle formed by the two radii and the chord. That triangle’s area is ½r²sin θ, which is where the sine term comes from.
How do I work out fluid depth in a round tank?
The cross-section is a segment. Because segment area is not proportional to depth, the volume in a horizontal cylinder rises fastest through the middle: which is why a half-full reading is not half the angle.
For the whole circle, see the circle calculator. For angle unit conversion, use the angle converter.