How to convert between degrees and radians?

Degrees to radians: rad = deg × π/180. Radians to degrees: deg = rad × 180/π. For example, 180° = π rad, 90° = π/2 rad.

How many degrees is 1 radian?

1 radian ≈ 57.2958°. 1 radian = 180/π degrees ≈ 57.29577951308232 degrees. Conversely, 1 degree = π/180 radians ≈ 0.01745 radians.

The gradian (grad/gon) is a unit of angular measurement commonly used in surveying. A full circle is 400 gradians, 1 gradian = 0.9 degrees = π/200 radians. Conversion: grad = deg × 10/9.

📐 Angle Converter

Convert between degrees (°), radians (rad), and gradians (grad)

Enter Value

Input Unit

Degree (°)180

Radian (rad)3.14159

Gradian (grad)200

* Conversion: 360° = 2π rad = 400 grad

* 1° = π/180 rad ≈ 0.01745 rad

* 1 rad = 180/π° ≈ 57.296°

* 1 grad = 0.9° = π/200 rad

Angle Converter Guide

This tool quickly converts between degrees (°), radians (rad), and gradians (grad). Enter a value and select the unit to automatically convert to other units.

Angle conversion is very common in mathematics, physics, and engineering, especially when working with trigonometric functions.

Angle Unit Conversion Reference

Conversion relationships: - 360° = 2π rad ≈ 6.2832 rad = 400 grad - 1° = π/180 rad ≈ 0.01745 rad = 10/9 grad - 1 rad = 180/π° ≈ 57.296° ≈ 63.662 grad - 1 grad = 0.9° = π/200 rad

Common angles: - 30° = π/6 rad ≈ 0.5236 rad - 45° = π/4 rad ≈ 0.7854 rad - 60° = π/3 rad ≈ 1.0472 rad - 90° = π/2 rad ≈ 1.5708 rad - 180° = π rad ≈ 3.1416 rad - 360° = 2π rad ≈ 6.2832 rad

Gradians are primarily used in surveying and civil engineering, dividing a right angle into 100 equal parts.

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Angle Converter Instructions

The angle converter supports conversion between degrees (°), radians (rad), and gradians (grad). Enter a value and select the source unit to see the equivalent value in all angle units.

An angle measures the amount of rotation or the opening between two rays. The most commonly used system in daily life is the degree system (360° = 1 full circle), while mathematics and physics more commonly use the radian system (2π rad = 1 full circle). This tool covers both systems as well as the more precise arcminute/arcsecond units, meeting precision needs from astronomical observation to mechanical machining.

Angle Unit Conversion Relationships

【Basic Relationships】
1 full circle = 360° = 2π rad ≈ 6.283 rad
       = 400 grad (gradians)
       = 21600' (arcminutes)
       = 1296000" (arcseconds)

【Degrees and Radians】
1° = π/180 rad ≈ 0.0174533 rad
1 rad = 180/π ° ≈ 57.2958°

【Arcminutes and Arcseconds】
1° = 60' (60 arcminutes)
1' = 60" (60 arcseconds)
1° = 3600"

【Gradians】
100 grad = 90° = π/2 rad (right angle)

Conversion Examples

📋

【Example 1】Common special angles 30° = π/6 ≈ 0.5236 rad = 33.333 grad 45° = π/4 ≈ 0.7854 rad = 50 grad 60° = π/3 ≈ 1.0472 rad = 66.667 grad 90° = π/2 ≈ 1.5708 rad = 100 grad 180° = π ≈ 3.1416 rad = 200 grad 【Example 2】Astronomical observation precision Moon apparent diameter: about 31' (arcminutes) = 0.5167° ≈ 0.00901 rad Sun apparent diameter: about 32' Both are nearly the same → total solar eclipses are possible! 【Example 3】Precision manufacturing High-precision machine tool: tolerance ±10" (arcseconds) = ±0.00278° ≈ ±4.85×10⁻⁵ rad GPS positioning accuracy: horizontal ±3m → geocentric angle about ±0.04" 【Example 4】Circular motion Uniform circular motion one revolution = 2π rad Frequency f(Hz) → Angular velocity ω = 2πf rad/s 3000rpm motor: ω = 2π×50 = 314.16 rad/s

Important Notes

💡

- Radians are dimensionless (length/length), but "rad" is often written in calculations for clarity - Gradians (grad) are used in surveying and engineering in some European countries, but are less common in China - Arcminutes/arcseconds are widely used in astronomy, navigation, and optics — for example, latitude/longitude is expressed in degrees-minutes-seconds format - JavaScript's Math.sin/cos default parameter is radians, not degrees - Small angle approximation: sin θ ≈ θ (when θ is in radians and very small)

Application Scenarios

- Mathematics education: Quick conversion between degrees and radians for trigonometric calculations
- Navigation and positioning: GPS coordinate degrees-minutes-seconds format conversion, heading angle calculation
- Astronomical observation: Angular measurement of celestial body positions, telescope pointing control
- Mechanical machining: Workpiece angle marking, CNC machine programming
- Game development: Character rotation angles, camera view angle radian/degree switching

Why Are There So Many Angle Units?

The origin of 360° traces back to the sexagesimal (base-60) system used by the ancient Babylonians — they observed that a year had approximately 360 days (actually 365.24), so they divided the circle into 360 parts. 60 is also a highly composite number (divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), making equal division calculations convenient.

The advantage of the radian system lies in its mathematical simplicity: lim(sin x/x) = 1 and d/dx(sin x) = cos x only hold true under the radian system. This is why advanced mathematical tools such as calculus and Fourier transforms are based on radians.

The gradian (grad) was a "decimalization" attempt proposed during the French Revolution — dividing a right angle into 100 grad instead of 90°. Although it failed to replace the degree system, it is still retained in surveying in some parts of Europe.

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