Angle Converter

Simplify your math by switching between degrees, radians, and gradians. Get precise results for geometry, engineering, and trigonometric calculations.

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How to Convert Angles?

To convert angles between different units, use the appropriate conversion factors. For example, to convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.

Angle Conversion Principles

1. Degrees (°)

The degree is the most common unit for measuring angles. A full circle is 360 degrees.

2. Radians (rad)

The radian is the SI unit for measuring angles. A full circle is 2π radians (approximately 6.28318 radians).

3. Gradians (grad)

The gradian divides a circle into 400 units. A full circle is 400 gradians.

4. Minutes and Seconds

Minutes (′) and seconds (″) are subdivisions of degrees. 1 degree = 60 minutes, 1 minute = 60 seconds.

5. Conversion Formulas

Degrees to Radians: ° × π/180 = rad
Radians to Degrees: rad × 180/π = °
Degrees to Gradians: ° × 400/360 = grad
Gradians to Degrees: grad × 360/400 = °
Degrees to Minutes: ° × 60 = ′
Minutes to Degrees: ′ ÷ 60 = °
Minutes to Seconds: ′ × 60 = ″
Seconds to Minutes: ″ ÷ 60 = ′
Degrees to Turns: ° ÷ 360 = turn
Turns to Degrees: turn × 360 = °

6. Common Angle Values

0° = 0 rad = 0 grad
90° = π/2 rad = 100 grad
180° = π rad = 200 grad
270° = 3π/2 rad = 300 grad
360° = 2π rad = 400 grad

Common Angle Units

Degrees (°)

The most widely used unit for measuring angles. A full circle is 360 degrees.

Radians (rad)

The SI unit for measuring angles. Used extensively in mathematics and physics.

Gradians (grad)

Also called 'gons', used in some surveying and engineering applications.

Minutes (′) and Seconds (″)

Subdivisions of degrees used for precise angle measurements.

Turns and Revolutions

Complete rotations around a circle. 1 turn = 1 revolution = 360° = 2π rad.

Real-World Usage Scenarios

Mechanical Engineering - Robotics - Designers often convert between turns and degrees when programming stepper motors or robotic joints. For instance, translating a motor's full revolution (1 turn) into specific degree increments ensures precise positioning in CNC machinery.
Civil Engineering - Land Surveying - Surveyors in many European regions use gradians (gons) rather than degrees. Converting legacy survey data from gradians to decimal degrees is a common step when importing topographical maps into modern GIS software.
Software Development - Physics Engines - Most programming libraries, such as Python's math module or JavaScript's Math object, require angles in radians for trigonometric functions. Developers must convert UI-facing degree values to radians before executing physics simulations or rotation logic.
Maritime Navigation - Cartography - High-precision navigation requires converting between degrees-minutes-seconds (DMS) and decimal degrees. This is essential for plotting coordinates on traditional nautical charts where one minute of latitude equals one nautical mile.

Frequently Asked Questions

Why do mathematicians prefer radians over degrees?

Radians are a 'natural' unit based on the radius of a circle. They simplify calculus operations; for example, the derivative of sin(x) is exactly cos(x) only when x is in radians. Using degrees would introduce a cumbersome factor of π/180 into every calculation.

What is the difference between a gradian and a degree?

A degree divides a full circle into 360 parts, while a gradian (or gon) divides it into 400. Gradians were designed to be a metric-friendly unit where a right angle is exactly 100 units, making it easier for surveyors to perform mental calculations.

How accurate is the conversion between units like arcseconds and radians?

The tool uses high-precision floating-point arithmetic to ensure that even tiny units like arcseconds (1/3600th of a degree) are converted accurately for applications in astronomy and precision optics.

When should I use turns or revolutions instead of degrees?

Turns and revolutions are best suited for mechanical engineering and rotational dynamics, such as calculating the RPM (revolutions per minute) of an engine or the total distance traveled by a wheel over several rotations.

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