What this calculator does
The perimeter of a sector is the distance right around the wedge-shaped piece of a circle, and it is easy to shortchange it by forgetting one of its three parts. There are the two straight radii bounding the wedge, plus the curved arc between them, and all three have to be added together, not just the arc on its own.
This calculator takes a radius and a central angle and returns that full perimeter of a sector directly, with the arc length, the two straight edges, and the sector area worked out alongside it, so the arc is never mistaken for the whole boundary.
The formula
Arc length is a fraction of the full circumference, in proportion to the angle out of 360°: arc = 2πr × θ/360. The two straight radii simply add 2r on top of that. Perimeter of a sector formula is therefore perimeter = 2r + 2πr × θ/360, the sum of both straight edges and the curved one.
| Term | Meaning |
|---|---|
| r | The radius: the length of each of the two straight edges of the sector. |
| θ | The central angle the sector spans, measured in degrees. |
| Arc length | The length of the curved edge only, a fraction of the full circle’s circumference. |
The inputs explained
| Field | What to enter |
|---|---|
| Radius | The radius of the circle the sector is cut from. |
| Central angle (°) | The central angle the sector spans, in degrees, from just above 0° up to 360° for the full circle. |
When to use it
Edging or trim around a pie-slice shaped panel
A pie-slice shaped garden bed, splashback panel or pizza-shaped sign needs a length of edging or trim equal to its full perimeter, the two straight sides plus the curved edge, not the arc alone.
Fencing a wedge-shaped block of land
A wedge-shaped block bounded by two straight property lines meeting at a point and a curved road frontage has a fencing length worked out the same way, as a sector’s perimeter.
How to find the perimeter of a sector for a geometry problem
A textbook question giving a radius and central angle is solved directly here, without separately working out the arc length and then remembering to add the two radii 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 the perimeter of a sector change with the angle?
A fixed radius of 10, across a range of central angles from a narrow wedge to a half circle.
| Central angle | Perimeter of the sector | Arc length |
|---|---|---|
| 30° | 25.236 | 5.236 |
| 60° | 30.472 | 10.472 |
| 90° | 35.708 | 15.708 |
| 180° | 51.416 | 31.416 |
| 270° | 67.124 | 47.124 |
Perimeter of a sector across a range of radii, angle fixed
A fixed central angle of 90°, a quarter circle, across a range of radius values.
| Radius | Perimeter of the sector | Sector area |
|---|---|---|
| 2 | 7.142 | 3.142 |
| 4 | 14.283 | 12.566 |
| 6 | 21.425 | 28.274 |
| 8 | 28.566 | 50.265 |
| 10 | 35.708 | 78.540 |
Questions
What is the perimeter of a sector formula?
Perimeter = 2r + 2πr × θ/360, where r is the radius and θ is the central angle in degrees. It is the two straight radii plus the curved arc length.
How to find the perimeter of a sector without the angle in degrees?
If the angle is in radians, arc length is simply r × θ rather than 2πr × θ/360, so perimeter becomes 2r + rθ. This calculator takes degrees, so convert radians to degrees first by multiplying by 180/π.
Is the perimeter of a sector the same as the arc length?
No. The arc length is only the curved part. The perimeter also includes the two straight radii bounding the sector, so it is always longer than the arc length alone, by exactly 2r.
How is this different from a sector area calculator?
Sector area measures the space enclosed inside the wedge, in square units, useful for material or coverage. Perimeter measures the distance around the boundary, in linear units, useful for edging, fencing or trim.
For the area of a sector as the headline figure instead, see the sector area calculator. For arc length, chord length and segment area together with a full circle sector breakdown, see the circle sector and arc calculator.