# Angle Converter

> Convert between angle units - degrees, radians, gradians, turns, arcminutes, arcseconds

**Category:** Conversion
**Keywords:** angle, degrees, radians, gradians, turns, arcminutes, arcseconds, milliradians, mrad, gon, conversion, trigonometry
**URL:** https://complete.tools/angle-converter

## How it calculates

All conversions use degrees as the intermediate base unit. When you enter a value in any unit, the tool first converts it to degrees, then converts from degrees to all other supported units. This two-step approach ensures consistent accuracy across all conversion pairs.

**Conversion Factors (to and from degrees):**
- 1 radian = 180/π ≈ 57.2958°
- 1 gradian = 0.9° (400 gradians in a full circle)
- 1 turn = 360° (1 complete rotation)
- 1 arcminute = 1/60° ≈ 0.0167°
- 1 arcsecond = 1/3600° ≈ 0.000278°
- 1 milliradian = 180/(1000π) ≈ 0.0573°

**Sample calculation:** Converting 90 degrees to radians uses the formula: radians = degrees × (π/180), so 90° × (π/180) = π/2 ≈ 1.5708 rad. Converting back: 1.5708 × (180/π) = 90°.

## Who should use this

Students learning trigonometry, calculus, or physics need this tool for converting between the degree-based angles in textbooks and the radian-based angles required by mathematical functions. The instant conversion eliminates arithmetic errors and saves time during homework and exams.

Engineers working with rotational mechanics, signal processing, or control systems frequently need to convert between radians (used in mathematical analysis) and degrees (used in specifications and documentation). CAD designers and architects also convert angles when working with different software systems or international standards.

Surveyors converting between different surveying angle systems benefit from quick gradian-to-degree conversions. Astronomers and navigators working with celestial coordinates use arcminute and arcsecond conversions for precise positional data. Military personnel convert between mils (milliradians) and degrees for targeting and ballistic calculations.

Programmers implementing graphics, games, or scientific applications need to convert angles because programming languages typically use radians while user interfaces display degrees. This tool helps verify calculations and debug angle-related code issues.

## Worked examples

**Trigonometry Homework:** A student needs to find sin(45°) using a calculator that requires radians. Convert 45° to radians: 45 × (π/180) = 0.7854 rad. Enter 0.7854 into the calculator to get sin(0.7854) = 0.7071.

**Navigation Calculation:** A ship's course is 127.5° from true north. A navigation computer requires gradians. Convert: 127.5° × (400/360) = 141.67 gradians.

**Astronomical Observation:** A telescope pointing system specifies an angle of 2.5 arcminutes. Convert to degrees for a star chart: 2.5 arcminutes × (1/60) = 0.0417°.

**Engineering Specification:** A motor shaft rotates 2.5 turns per second. Convert to degrees per second: 2.5 × 360 = 900°/s. Convert to radians per second: 2.5 × 2π = 15.71 rad/s.

**Military Ranging:** A spotter measures an angle of 50 milliradians. Convert to degrees: 50 × 0.0573 = 2.865°.

## Limitations

The converter uses mathematical approximations of π that provide accuracy to many decimal places but are not infinitely precise. For most practical applications, this precision far exceeds measurement accuracy, but theoretical calculations requiring exact values should use symbolic mathematics instead.

The tool converts single angle values rather than angular velocities or angular accelerations, which have their own unit systems. It does not perform trigonometric calculations or solve geometry problems, focusing solely on unit conversion between different angle measurement systems.

Very large or very small input values may display in scientific notation for readability. The converter assumes input values are valid numbers and does not normalize angles to standard ranges (such as 0-360° or -180° to 180°).

## FAQs

**Q:** Why do mathematicians prefer radians over degrees?


**A:** Radians create simpler formulas in calculus and physics. The derivative of sin(x) is cos(x) only when x is in radians. Degrees require additional conversion factors in these formulas, making radians the natural choice for mathematical analysis.


**Q:** What is the difference between arcminutes and milliradians?


**A:** Arcminutes divide a degree into 60 parts (used in astronomy and navigation), while milliradians divide a radian into 1000 parts (used in military and engineering applications). One arcminute is approximately 0.29 milliradians.


**Q:** Why do surveyors use gradians?


**A:** Gradians divide a right angle into exactly 100 parts, making percentage calculations intuitive. A 10% grade equals exactly 10 gradians, which simplifies slope calculations in civil engineering and surveying work.


**Q:** How accurate are these conversions?


**A:** The conversions use double-precision floating-point arithmetic, providing accuracy to approximately 15 significant digits, far exceeding the precision of typical measurements.

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