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Angle Converter

Convert angle measurements between degrees, radians, gradians, arcminutes, arcseconds, and turns (revolutions). Essential for trigonometry, navigation, surveying, astronomy, and any application requiring precise angular unit conversions across different systems.

Reviewed by Chase FloiedUpdated

This free online angle converter provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

The angle to convert

Source angle unit

Target angle unit

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Angle Converter. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Angle Converter instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Angle Converter Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Angle Converter when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
  • •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
  • •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
  • •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.

About This Calculator

The Angle Converter is a free, browser-based calculation tool for engineers, students, and technical professionals. Convert angle measurements between degrees, radians, gradians, arcminutes, arcseconds, and turns (revolutions). Essential for trigonometry, navigation, surveying, astronomy, and any application requiring precise angular unit conversions across different systems. It implements standard formulas and supports both metric (SI) and imperial unit systems with automatic unit conversion. All calculations are performed instantly in your browser with no data sent to a server. Use this calculator as a quick reference and sanity-check tool during design, analysis, and learning. Always verify results against primary engineering references and applicable standards for any safety-critical application.

About Angle Converter

The Angle Converter provides precise conversions between the most widely used angular measurement units. Degrees are the everyday standard for navigation, construction, and geometry. Radians are the natural mathematical unit used in calculus and physics. Gradians (gons) divide a right angle into 100 parts for surveying convenience. Arcminutes and arcseconds subdivide degrees for high-precision work in astronomy and geodesy. Turns (revolutions) express full rotations. This tool converts seamlessly among all six units, making it invaluable for students, engineers, astronomers, and surveyors.

The Math Behind It

An angle measures the rotation between two rays sharing a common endpoint (vertex). The choice of unit depends on the application, each offering advantages in different contexts. Degrees divide a full rotation into 360 equal parts. This system dates to ancient Babylon and may relate to the approximate number of days in a year. It is the most widely recognized angular unit, used in navigation, construction, everyday geometry, and most consumer instruments. Radians define angle as the ratio of arc length to radius: one radian is the angle subtended by an arc equal in length to the radius. A full circle is 2*pi radians (approximately 6.28318). Radians are the natural unit for calculus because the derivatives of trigonometric functions take their simplest form: d/dx(sin x) = cos x only when x is in radians. All scientific computing libraries use radians as the default. Gradians (also called gons or grads) divide a right angle into exactly 100 parts, making a full circle 400 gradians. This decimal system simplifies mental arithmetic in surveying: north is 0 grad, east is 100, south is 200, west is 300. Many surveying instruments offer a gradian mode alongside degrees. Arcminutes and arcseconds subdivide degrees in a sexagesimal (base-60) system: 1 degree = 60 arcminutes, 1 arcminute = 60 arcseconds. This provides extremely fine angular resolution. Astronomers use arcseconds to measure the apparent size and separation of celestial objects. GPS coordinates often express latitude and longitude in degrees, arcminutes, and arcseconds (DMS format). Turns (revolutions) express angle as a fraction of a full rotation. One turn = 360 degrees = 2*pi radians. Turns are intuitive for describing rotating machinery: a motor speed of 3600 RPM means 3600 turns per minute. The tau (tau = 2*pi) movement advocates using turns as the fundamental angular unit, arguing that it simplifies many formulas.

Formula Reference

Angle Conversion via Degrees

result = value * toDeg[from] * fromDeg[to]

Variables: value = input angle, toDeg = factor to degrees, fromDeg = factor from degrees

Worked Examples

Example 1: Right Angle: Degrees to Radians

Convert 90 degrees to radians.

Step 1:Apply formula: radians = degrees * pi / 180
Step 2:radians = 90 * 3.14159 / 180
Step 3:radians = 1.5708

90 degrees equals pi/2 radians, approximately 1.5708.

Example 2: Navigation: Degrees to Gradians

Convert a 45-degree bearing to gradians.

Step 1:Apply conversion: gradians = degrees / 0.9
Step 2:gradians = 45 / 0.9 = 50

45 degrees equals 50 gradians.

Common Mistakes & Tips

  • !Using degrees in trigonometric functions that expect radians. Most programming languages (JavaScript, Python, C) use radians. Passing 90 instead of pi/2 to sin() gives sin(90 radians) which is approximately 0.894, not 1.
  • !Confusing gradians with degrees. A right angle is 90 degrees but 100 gradians. Using one for the other causes 11% error in all calculations.
  • !Forgetting that arcminutes and arcseconds are fractions of degrees, not radians. One arcminute is 1/60 of a degree, or about 0.000291 radians.

Related Concepts

Torque Converter

Torque calculations involving angular displacement require consistent angle units. Convert between torque units for rotational mechanics.

Frequency Converter

Angular frequency (rad/s) relates to turns and degrees per second. Convert frequency units for rotational and oscillatory systems.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Frequency Converter

Torque Converter

Frequently Asked Questions

Why are there 360 degrees in a circle?

The number 360 likely originated in ancient Babylon, where a base-60 number system was used. It may relate to the approximately 365 days in a year, rounded to 360 for mathematical convenience. The number 360 has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360), making it highly practical for dividing circles into equal parts.

What is the difference between a radian and a degree?

A degree is 1/360 of a full rotation, while a radian is the angle subtended by an arc equal to the radius. One radian is approximately 57.296 degrees. Radians are preferred in mathematics because they make formulas simpler; for example, the arc length formula is simply s = r * theta when theta is in radians.

When should I use gradians?

Gradians are primarily used in surveying and some European engineering contexts. Their decimal subdivision (100 gons per right angle) makes bearing calculations simpler than the sexagesimal degree system. Most theodolites and total stations offer a gradian mode, and French and some European cadastral surveys historically used gradians.