# Конвертер углов: градусы, радианы, DMS и грады

> Переводите десятичные градусы, градусы-минуты-секунды (DMS), радианы, грады, обороты, угловые минуты и секунды и миллирадианы.

Интерактивная версия: https://www.calcopenly.com/ru/geometry/angle-dms-converter
Тема: Геометрические калькуляторы

Type an angle in any of seven units and it is converted to all the others. Degrees accept decimals or degrees, minutes and seconds: 1° = 60′ = 3,600″, so 12°30′15″ is 12 + 30/60 + 15/3600 = 12.5041667°. Radians use 180° = π rad, gradians 400 to a full turn and milliradians 1,000 to a radian. When the angle is a simple fraction of π, the π form is shown too, such as 5π/72 for 12.5°.

The default, 12°30′15″, is 0.2182389 rad or 45,015″. GPS coordinates and survey bearings are the usual case: a latitude written 33°51′35″ S becomes −33.8597222° for a spreadsheet or mapping tool, with south and west negative.

Only the last DMS part may carry decimals, and minutes and seconds must be below 60. Seconds are rounded to four decimal places, about 3 mm of latitude on the ground.

## Входные данные

- **Угол**: Degrees accept DMS such as 12°30'15", 12 30 15 or 33°51'35" S; radians accept expressions such as pi/4
- **Unit of the angle** (варианты: Degrees (decimal or DMS), Радианы, Gradians (gon), Turns, Arcminutes, Arcseconds, Milliradians)

## Результаты

- Градусы (°) — основной результат
- Degrees, minutes, seconds
- Радианы (rad)
- Radians as a multiple of π
- Грады (grad)
- Turns
- Arcminutes (′)
- Arcseconds (″)
- Milliradians (mrad)
- Same direction within 0°–360° (°)

## Формула

$$
1^\circ = 60' = 3600'' = \frac{\pi}{180}\,\text{rad} = \frac{10}{9}\,\text{grad} = \frac{1}{360}\,\text{turn}
$$

## Примеры с решением

### 12°30'15" (DMS)

- Угол: 12°30'15"
- Unit of the angle: Degrees (decimal or DMS)
- **Градусы: 12.5041666667 °**
- **Радианы: 0.2182388786 rad**
- **Грады: 13.8935185185 grad**
- **Degrees, minutes, seconds: 12° 30′ 15″**
- **Arcseconds: 45,015 ″**
- Источник проверки: Python 3.8 fractions: 12 + 30/60 + 15/3600 = 3001/240; math.radians of that; ×10/9 for gradians

### 1 radian

- Угол: 1
- Unit of the angle: Радианы
- **Градусы: 57.2957795131 °**
- **Degrees, minutes, seconds: 57° 17′ 44.8062″**
- **Turns: 0.1591549431**
- Источник проверки: Python 3.8 math: degrees(1) = 57.29577951308232; 0.2957795…×60 = 17.7467707…′, 0.7467707…×60 = 44.80625″; 1/(2π)

### π radians typed as an expression

- Угол: pi
- Unit of the angle: Радианы
- **Градусы: 180 °**
- **Radians as a multiple of π: π**
- **Грады: 200 grad**
- Источник проверки: π rad = 180° = 200 grad by definition

### 400 gradians (full turn, edge case)

- Угол: 400
- Unit of the angle: Gradians (gon)
- **Градусы: 360 °**
- **Turns: 1**
- **Same direction within 0°–360°: 0 °**
- **Degrees, minutes, seconds: 360° 0′ 0″**
- Источник проверки: 400 grad = 1 turn = 360° by definition

### Minus half a degree written as −0°30' (sign edge case)

- Угол: -0°30'
- Unit of the angle: Degrees (decimal or DMS)
- **Градусы: -0.5 °**
- **Degrees, minutes, seconds: −0° 30′ 0″**
- **Arcminutes: -30 ′**
- **Same direction within 0°–360°: 359.5 °**
- Источник проверки: The sign applies to the whole angle: −(0 + 30/60)

### Latitude 33°51'35" S

- Угол: 33°51'35" S
- Unit of the angle: Degrees (decimal or DMS)
- **Градусы: -33.8597222222 °**
- Источник проверки: Python 3.8 fractions: −(33 + 51/60 + 35/3600); south latitudes are negative

## Вопросы

### How do you convert degrees, minutes and seconds to decimal degrees?

Add the minutes divided by 60 and the seconds divided by 3,600 to the whole degrees: 12°30′15″ = 12 + 0.5 + 0.0041667 = 12.5041667°. For a south latitude or a west longitude, make the result negative, so 33°51′35″ S is −33.8597222°. The sign applies to the whole angle, not only to the degrees.

### How do you convert decimal degrees to degrees, minutes and seconds?

Keep the whole number as degrees, multiply the fractional part by 60 for minutes, then multiply the new fractional part by 60 for seconds. For 57.2957795°: 0.2957795 × 60 = 17.7467707′, and 0.7467707 × 60 = 44.806″, giving 57°17′44.806″. That angle is one radian.

### Сколько градусов в радиане?

One radian is 180/π ≈ 57.2958°, or 57°17′44.8″. It is the angle at the centre of a circle cut off by an arc as long as the radius, so a full turn is 2π ≈ 6.2832 rad. The radian is the SI unit of plane angle; the degree, minute and second are accepted for use with the SI (BIPM SI Brochure, Table 8).

### Что такое град?

A gradian, or gon, is 1/400 of a full turn, so a right angle is exactly 100 grad and 1 grad = 0.9°. It is used mainly in surveying in parts of continental Europe, where splitting the right angle into 100 parts suits decimal arithmetic. 200 grad equals 180°, or π rad.

### What is a milliradian?

A milliradian (mrad) is one thousandth of a radian, about 0.0573° or 3.44 arcminutes. At a distance of 1,000 m it spans about 1 m, which is why rifle scopes use it for holdover and range estimates. The NATO mil used for artillery is a rounded version: 6,400 to a circle, against 6,283.2 true milliradians.

### Насколько точен «Конвертер углов: градусы, радианы, DMS и грады»?

Точность зависит от введённых данных и допущений метода. Десятичная арифметика использует 50 значащих цифр, но оценки, численные методы и исходные данные могут быть менее точными; округление на экране не устраняет эти ограничения. Решённые примеры, проверенные по независимым источникам: 8. Например, «12°30'15" (DMS)» проверяется по источнику Python 3.8 fractions: 12 + 30/60 + 15/3600 = 3001/240; math.radians of that; ×10/9 for gradians.

### Откуда взята методика?

BIPM, The International System of Units (SI brochure, 9th ed.), Table 8 — degree, minute and second of arc; NIST SP 811 (2008), Appendix B.9 — factors for plane angle.

## Источники

- [BIPM, The International System of Units (SI brochure, 9th ed.), Table 8 — degree, minute and second of arc](https://www.bipm.org/en/publications/si-brochure)
- [NIST SP 811 (2008), Appendix B.9 — factors for plane angle](https://www.nist.gov/pml/special-publication-811)