# Angle converter

> Convert degrees to radians and radians to degrees, plus gradians, arcminutes, arcseconds, turns and milliradians, with π carried to 50 digits.

Interactive version: https://www.calcopenly.com/conversion/angle-converter
Subject: Unit converters

The radian is the SI unit of angle: the angle whose arc equals the radius, so a full turn is 2π rad. Degrees divide the turn into 360 and gradians (gon) into 400, and arcminutes and arcseconds are 1/60 and 1/3,600 of a degree. Converting degrees to radians multiplies by π/180, so the result is irrational and is shown to 12 significant figures.

The default converts a right angle: 90° = π/2 = 1.57079632679 rad. Programmers need this because the trigonometric functions of most languages take radians, and surveyors and astronomers move between degrees, gradians and arcseconds.

Conversions among degrees, arcminutes, arcseconds, gradians and turns never involve π and are exact: 1 turn = 360° = 400 gon. The milliradian here is the true 0.001 rad, about 6,283.2 to a turn.

## Inputs

- **Value**: A number in the “from” unit, or with its own unit
- **From** (options: radian (rad), degree (°), gradian (gon), arcminute (′), arcsecond (″), turn, milliradian (mrad))
- **To** (options: radian (rad), degree (°), gradian (gon), arcminute (′), arcsecond (″), turn, milliradian (mrad))

## Results

- Converted value — main result
- Conversion
- Conversion factor
- Exactness

## Formula

$$
\begin{gathered} x_{\text{to}} = x_{\text{from}} \times \dfrac{f_{\text{from}}}{f_{\text{to}}} \\ f = \text{one unit expressed in } \text{rad} \end{gathered}
$$

## Worked examples

### 90° in radians

- Value: 90 °
- From: degree (°)
- To: radian (rad)
- **Converted value: 1.5707963268 rad**
- **Exactness: Exact definitions, involves π**
- Checked against: 90 × π/180 = π/2 = 1.57079632679489661923… (π to 60 digits in Python decimal)

### 1 turn in degrees

- Value: 1 turn
- From: turn
- To: degree (°)
- **Converted value: 360 °**
- **Exactness: Exact**
- Checked against: 1 turn = 2π rad = 360° (π cancels)

### 1° in arcseconds

- Value: 1 °
- From: degree (°)
- To: arcsecond (″)
- **Converted value: 3,600 ″**
- Checked against: 1° = 60′ = 3600″ (SI Brochure Table 8)

### 1 radian in degrees

- Value: 1 rad
- From: radian (rad)
- To: degree (°)
- **Converted value: 57.2957795131 °**
- Checked against: 180/π = 57.29577951308232087… (Python decimal, 60 digits)

### 100 gradians in degrees

- Value: 100 gon
- From: gradian (gon)
- To: degree (°)
- **Converted value: 90 °**
- Checked against: 400 gon = 360°, so 100 gon = 90°

## Questions

### How do I convert degrees to radians?

Multiply by π/180, about 0.0174533. 90° is π/2 = 1.5708 rad, 180° is π = 3.14159 rad and 45° is π/4 = 0.785398 rad. To go back, multiply radians by 180/π, so 1 rad is 57.2958°. JavaScript's Math.sin and Python's math.sin both expect radians.

### How many degrees are in a radian?

57.2957795°, which is 180/π. A radian is the angle at the centre of a circle cut off by an arc as long as the radius; a full circumference is 2π radii, so a full turn is 2π rad = 360°. The SI Brochure defines the radian as 1 m/m, a coherent SI unit.

### What is a gradian?

One 400th of a full turn, so a right angle is exactly 100 gradians (symbol gon, also called grads). One gon is 0.9° or 54 arcminutes. It is used mainly in surveying, where a quarter circle of 100 keeps slopes and bearings decimal, and calculators label the mode GRAD.

### Is a mil the same as a milliradian?

Not always. A true milliradian is 0.001 rad, about 6,283.2 per turn, and that is the unit used here. NATO compasses and artillery use a mil of 1/6,400 turn, 0.98175 mrad, chosen so the circle divides evenly. Rifle scopes marked MRAD use the true 0.001 rad, where 1 mrad spans 10 cm at 100 m.

### How many arcseconds are in a degree?

3,600: a degree is 60 arcminutes (′) and an arcminute is 60 arcseconds (″), as listed in Table 8 of the SI Brochure. One arcsecond of latitude is about 30.9 m on the ground, because the nautical mile, 1,852 m, was based on one arcminute of latitude.

### How accurate is the angle converter?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “90° in radians” is checked against 90 × π/180 = π/2 = 1.57079632679489661923… (π to 60 digits in Python decimal).

### Where does the method come from?

NIST SP 811 (2008), Guide for the Use of the SI, Appendix B — conversion factors; NIST Handbook 44, Appendix C — General tables of units of measurement; BIPM SI Brochure, 9th edition (2019), Table 8 — degree, minute, second of arc.

## Sources

- [NIST SP 811 (2008), Guide for the Use of the SI, Appendix B — conversion factors](https://www.nist.gov/pml/special-publication-811)
- [NIST Handbook 44, Appendix C — General tables of units of measurement](https://www.nist.gov/pml/owm/nist-handbook-44-current-edition)
- [BIPM SI Brochure, 9th edition (2019), Table 8 — degree, minute, second of arc](https://www.bipm.org/en/publications/si-brochure)