# Angle Converter

> Converts plane angles between degrees, radians, milliradians, gradians (gon), arcminutes, arcseconds and turns (revolutions) using π rad = 180° = 200 gon = 0.5 turn.

- Calculator id: `angle` · Category: Unit Conversion (`conversion`) · Tool name: `convert_angle`
- Canonical page: https://tttkmbb.com/conversion/angle · This document: https://tttkmbb.com/conversion/angle.md · JSON definition: https://tttkmbb.com/conversion/angle.json

## Purpose

Converts plane angles between degrees, radians, milliradians, gradians (gon), arcminutes, arcseconds and turns (revolutions) using π rad = 180° = 200 gon = 0.5 turn.

**Use when:** You need an angle in another unit, e.g. degrees to radians for trigonometry, arcseconds for astronomy or turns for rotation counts.

**Do not use when:** You need an angle from a slope or triangle (use slope, right-triangle or triangle-solver) or a rotational speed (use frequency for rpm and rad/s).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `value` | number |  | required | Amount to convert, expressed in from_unit. |
| `from_unit` | enum: deg \| rad \| mrad \| grad \| arcmin \| arcsec \| turn |  | required | Unit of the input value. Accepted symbols: deg, rad, mrad, grad, arcmin, arcsec, turn; spelled-out names and common abbreviations are accepted too. |
| `to_unit` | enum: deg \| rad \| mrad \| grad \| arcmin \| arcsec \| turn |  | required | Unit to convert into (same symbols as from_unit). |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `result` | number |  | Converted value expressed in to_unit (rounded to 6 decimals; see result_text for very small or very large values). |
| `result_text` | string |  | The conversion as text with 6 significant figures, e.g. "5 mi = 8.04672 km". |
| `formula` | string |  | Relation used, e.g. "1 mi = 1.609344 km". |
| `factor` | number |  | Multiplier from from_unit to to_unit (result = value × factor); absent for non-linear conversions. |
| `conversion_table` | object |  | The input value expressed in every supported angle unit (6 significant figures), keyed by unit symbol. |

## Formula

`result = value × factor, factor = (1 from_unit in rad) / (1 to_unit in rad)`

## Data Sources

- NIST Special Publication 811 – Guide for the Use of the International System of Units (SI), Appendix B conversion factors — https://www.nist.gov/pml/special-publication-811 (standard, retrieved 2026-09-23)
- BIPM – The International System of Units (SI Brochure, 9th edition) — https://www.bipm.org/en/publications/si-brochure (standard, retrieved 2026-09-23)
- Wikipedia – Conversion of units — https://en.wikipedia.org/wiki/Conversion_of_units (reference, retrieved 2026-09-23)

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/angle?value=…&from_unit=…&to_unit=…`
- `POST https://tttkmbb.com/api/v1/calculate/angle` 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/angle · OpenAPI operationId `convert_angle` 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": "angle", "inputs": {…}}`

## Example

- 180 degrees to radians: inputs `{"value":180,"from_unit":"deg","to_unit":"rad"}` → `{"result":3.141593,"factor":0.0174532925,"formula":"1 ° = 0.01745329252 rad","conversion_table":{"grad":200,"turn":0.5,"arcmin":10800}}`
- 1 radian to degrees: inputs `{"value":1,"from_unit":"rad","to_unit":"deg"}` → `{"result":57.29578,"conversion_table":{"grad":63.662,"arcsec":206265}}`

```
GET https://tttkmbb.com/api/v1/calculate/angle?value=180&from_unit=deg&to_unit=rad
```

## Limitations

You need an angle from a slope or triangle (use slope, right-triangle or triangle-solver) or a rotational speed (use frequency for rpm and rad/s). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**How many degrees is one radian?**

180/π = 57.29578°; a full turn is 2π rad = 360° = 400 gon.

**What is a milliradian?**

1/1000 of a radian (0.0573°), used in optics and ballistics. The NATO 'mil' (1/6400 turn = 0.98175 mrad) is a different unit and is not included.

## Related

- [Right Triangle Calculator](https://tttkmbb.com/geometry/right-triangle.md) — Angles from triangle sides.
- [Slope Calculator](https://tttkmbb.com/math/slope.md) — Angle of a line from its gradient.
- [Frequency Converter](https://tttkmbb.com/conversion/frequency.md) — Rotational speed in rpm or rad/s.