# Circle Sector and Segment Calculator

> Computes, for a sector of a circle defined by its radius and central angle, the arc length, chord length, sector area, circular segment area and height, and the perimeters of the sector and segment.

- Calculator id: `circle-sector` · Category: Math (`math`) · Tool name: `calculate_circle_sector`
- Canonical page: https://tttkmbb.com/math/circle-sector · This document: https://tttkmbb.com/math/circle-sector.md · JSON definition: https://tttkmbb.com/math/circle-sector.json

## Purpose

Computes, for a sector of a circle defined by its radius and central angle, the arc length, chord length, sector area, circular segment area and height, and the perimeters of the sector and segment.

**Use when:** You know a circle's radius and a central angle (degrees or radians) and need the arc, chord, sector or segment measurements.

**Do not use when:** You need whole-circle properties from radius, diameter or circumference (use circle), or a regular polygon inscribed in a circle (use regular-polygon).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `radius` | number |  | required | Radius of the circle (any length unit; areas are in that unit squared). (> 0, max 1000000000) |
| `angle` | number |  | required | Central angle of the sector, at most 360° (2π rad). (> 0) |
| `unit` | enum: degrees \| radians |  | optional, default "degrees" | Unit of the central angle. |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `arc_length` | number |  | r·θ (θ in radians). |
| `chord_length` | number |  | 2·r·sin(θ/2): straight line between the arc's end points. |
| `sector_area` | number |  | r²·θ/2 (the 'pizza slice'). |
| `segment_area` | number |  | r²·(θ − sin θ)/2: area between the chord and the arc. |
| `segment_height` | number |  | r·(1 − cos(θ/2)): distance from the chord to the arc. |
| `sector_perimeter` | number |  | 2·r + arc length. |
| `segment_perimeter` | number |  | arc length + chord length. |
| `angle_degrees` | number | ° | The central angle in degrees. |
| `angle_radians` | number | rad | The central angle in radians. |
| `fraction_of_circle` | number |  | θ / 2π: the share of the full circle covered by the sector. |

## Formula

`θ = angle × π/180 (degrees); arc = r·θ; chord = 2r·sin(θ/2); sector_area = r²·θ/2; segment_area = r²·(θ − sin θ)/2; sagitta = r·(1 − cos(θ/2)); sector_perimeter = 2r + arc; segment_perimeter = arc + chord`

## Data Sources

- Wikipedia – Circular sector — https://en.wikipedia.org/wiki/Circular_sector (reference, retrieved 2026-09-24)
- Wikipedia – Circular segment — https://en.wikipedia.org/wiki/Circular_segment (reference, retrieved 2026-09-24)
- Wolfram MathWorld – Circular Sector — https://mathworld.wolfram.com/CircularSector.html (reference, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/circle-sector?radius=…&angle=…`
- `POST https://tttkmbb.com/api/v1/calculate/circle-sector` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/circle-sector · OpenAPI operationId `calculate_circle_sector` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "circle-sector", "inputs": {…}}`

## Example

- r = 10, 90°: inputs `{"radius":10,"angle":90,"unit":"degrees"}` → `{"arc_length":15.707963,"chord_length":14.142136,"sector_area":78.539816,"segment_area":28.539816,"segment_height":2.928932,"sector_perimeter":35.707963,"segment_perimeter":29.850099,"angle_radians":1.570796,"fraction_of_circle":0.25}`
- r = 6, 60°: inputs `{"radius":6,"angle":60}` → `{"arc_length":6.283185,"chord_length":6,"sector_area":18.849556,"segment_area":3.2611,"segment_height":0.803848,"fraction_of_circle":0.166667}`

```
GET https://tttkmbb.com/api/v1/calculate/circle-sector?radius=10&angle=90&unit=degrees
```

## Limitations

You need whole-circle properties from radius, diameter or circumference (use circle), or a regular polygon inscribed in a circle (use regular-polygon). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**What is the difference between a sector and a segment?**

A sector is bounded by two radii and the arc (like a pizza slice); a segment is the part of the sector beyond the chord, i.e. sector minus the isosceles triangle formed by the two radii.

**Can the angle exceed 180°?**

Yes, up to 360°. For angles above 180° the segment is the major segment (larger than a half disc) and the formulas still hold because sin θ becomes negative.

## Related

- [Circle Calculator](https://tttkmbb.com/geometry/circle.md) — Area and circumference of the whole circle.
- [Angle Converter](https://tttkmbb.com/conversion/angle.md) — Convert the central angle between degrees and radians.
- [Regular Polygon Calculator](https://tttkmbb.com/geometry/regular-polygon.md) — Polygons inscribed in or circumscribed about a circle.