What this calculator does
A sector is the pie-slice-shaped wedge of a circle bounded by two radii and the arc between them, and its area is a straightforward fraction of the whole circle, in direct proportion to the angle it spans out of 360°.
This sector area calculator leads with that headline figure, taking a radius and a central angle and returning the area of a sector directly, with the arc length, the fraction of the full circle, and the full circle area shown alongside it for context.
The formula
The sector area formula is area = πr²·θ/360, where θ is the central angle in degrees: the full circle area πr² scaled down to whatever fraction of 360° the angle represents. Arc length follows the same logic applied to the circumference instead: arc = 2πr·θ/360.
| Term | Meaning |
|---|---|
| r | Radius: the distance from the centre to the edge of the circle. |
| θ | Central angle: the angle at the centre spanning the sector, in degrees. |
| Sector | The wedge-shaped region bounded by two radii and the arc between them. |
The inputs explained
| Field | What to enter |
|---|---|
| Radius | The radius of the circle the sector is cut from. |
| Central angle (°) | The central angle spanning the sector, in degrees, between 0° and 360°. |
When to use it
A pie chart or wedge-shaped panel
A slice of a pie chart, a wedge of pizza, or a fan-shaped panel are all sectors, and the area of a sector formula gives the space each wedge covers from just its angle and radius.
A sprinkler or floodlight coverage area
A rotating sprinkler or a floodlight covering less than a full circle illuminates or waters a sector-shaped area, useful for working out coverage against a garden bed or car park.
Checking an area of sector formula answer
A geometry problem asking for sector area from a radius and angle is confirmed here directly, without separately working out what fraction of 360° the angle represents by hand.
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.
How does sector area change with the angle?
A fixed radius of 10, across a range of central angle values from a narrow wedge to a full circle.
| Central angle | Sector area | Arc length |
|---|---|---|
| 30° | 26.180 | 5.236 |
| 60° | 52.360 | 10.472 |
| 90° | 78.540 | 15.708 |
| 180° | 157.080 | 31.416 |
| 270° | 235.619 | 47.124 |
| 360° | 314.159 | 62.832 |
How does sector area change with the radius?
A fixed quarter-circle angle of 90°, across a range of radius values.
Questions
What is the area of a sector formula?
Area = πr²·θ/360, where r is the radius and θ is the central angle in degrees. It is the full circle area πr² scaled down by the fraction of 360° the angle covers.
What is the difference between a sector and a segment?
A sector is the full pie-slice, bounded by two radii and the arc. A segment is only the region between the arc and the straight chord joining its endpoints, the sector with the triangular part cut away.
What is the sector area formula in radians instead of degrees?
Area = ½r²θ, where θ is in radians. This calculator works in degrees and converts internally, but the two formulas describe the same relationship.
How is arc length related to sector area?
Both scale with the same fraction of the circle: arc length is that fraction of the circumference, and sector area is that fraction of the full circle area, so doubling the angle doubles both.
For arc length, chord and segment area together in one calculator, see the full circle sector and arc calculator. For the full circle’s own area, see the area of a circle calculator.