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Walk-through

How to Use This Calculator

3 steps

1

Enter the angle value

Type the number you want to convert into the Value field. The default of 90 shows a right angle, but any positive or negative number works.

2

Pick the from and to units

Choose the unit you're converting from — Degrees, Radians, Gradians, Arcminutes, Arcseconds, or Revolutions — and the unit you want to convert to. Use the swap button to flip them instantly.

3

Read the result

The converted value appears immediately. When converting to radians, an exact π fraction (like π/2 or 3π/4) is shown whenever the angle lands on a clean fraction of a circle, alongside the decimal value.

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Reference

Formula & Methodology

2 formulas

Convert through degrees

degrees = value × (unit's degree factor); result = degrees ÷ (target unit's degree factor)

Every angle unit converts through a common base — degrees. A degree factor tells you how many degrees equal one of that unit: 1 for degrees, 180/π for radians, 0.9 for gradians, 1/60 for arcminutes, 1/3600 for arcseconds, and 360 for revolutions. Multiplying the input by its factor gives degrees; dividing by the target unit's factor gives the final answer.

Exact π form

radians = degrees × π/180

A radian measure is exact when degrees/180 reduces to a simple fraction p/q with a small denominator (up to 24). The converter shows that reduced fraction as p·π/q instead of a rounded decimal whenever one exists.

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Glossary

Key Terms Explained

6 terms

Degree The most common angle unit, where a full circle is 360°. A right angle is 90°.

Radian The SI unit of angle, defined so a full circle equals 2π radians (about 6.2832 rad). One radian is about 57.2958°.

Gradian Also called a gon, a unit that divides a full circle into 400 equal parts, so a right angle is exactly 100 gradians. Used in some surveying and engineering fields.

Arcminute One-sixtieth of a degree, written with a prime symbol (′). Used in astronomy and navigation for fine angular measurement.

Arcsecond One-sixtieth of an arcminute (1/3600 of a degree), written with a double-prime symbol (″). Used for very precise angular measurements, such as star positions.

Revolution One full turn around a circle, equal to 360° or 2π radians. Also called a turn or cycle.

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Scenarios

Real-World Examples

3 worked examples

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Right angle

180° to radians

Value 180 From → To Degrees → Radians

180° equals exactly π radians (about 3.1416 rad) — the converter shows the exact π form because 180° is a clean fraction of a full circle.

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Full turn

1 revolution to degrees

Value 1 From → To Revolutions → Degrees

1 revolution equals 360° — a full trip around the circle back to the starting point.

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Fine measurement

1° to arcminutes

Value 1 From → To Degrees → Arcminutes

1° equals 60 arcminutes — useful when working with astronomical coordinates or precise navigation bearings.

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Deep Dive

Understanding the Angle Converter

3 chapters · 3 min read

Angles show up in different units depending on the field: degrees in everyday geometry, radians in calculus and physics, gradians in surveying, and arcminutes or arcseconds in astronomy and navigation. This calculator converts between all six units instantly and shows the exact π fraction for radian results whenever one exists.

How the conversion works

Every angle unit has a fixed relationship to a full circle: 360 degrees, 2π radians, 400 gradians, 21,600 arcminutes, or 1,296,000 arcseconds. The calculator converts your input to degrees first, then divides by the target unit's degree equivalent. This two-step approach (through a common base unit) avoids needing a separate conversion factor for every possible pair of units.

Why exact π form matters

A decimal radian value like 1.5708 is an approximation — the true value of π/2 radians is irrational and never terminates. For angles that are clean fractions of a circle (30°, 45°, 60°, 90°, and their multiples), expressing the result as a fraction of π (π/6, π/4, π/3, π/2) is both more precise and easier to work with in trigonometry, calculus, and physics formulas. The calculator checks denominators up to 24 and shows the exact form whenever one is found.

When to use each unit

Degrees are the everyday default for geometry, navigation headings, and most protractor-based measurements. Radians are the natural unit in calculus and physics — trigonometric derivatives, angular velocity, and arc length formulas all assume radians. Gradians appear in some European surveying and engineering traditions where a right angle equaling exactly 100 units simplifies certain calculations. Arcminutes and arcseconds are reserved for very fine angular measurements, such as star coordinates in astronomy or precise bearings in navigation and land surveying.

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Questions

Frequently Asked Questions

6 questions

How do I convert degrees to radians?+

Multiply the degree value by π/180 (approximately 0.017453). For example, 90° × π/180 = π/2 radians, which is about 1.5708 rad. The converter does this automatically and also shows the exact π fraction when one exists.

What is a gradian and how does it relate to degrees?+

A gradian (also called a gon) divides a full circle into 400 equal parts instead of 360, so a right angle is exactly 100 gradians. To convert degrees to gradians, multiply by 10/9 (or divide by 0.9). Gradians are mostly used in surveying and some European engineering contexts.

How many arcminutes are in a degree?+

There are 60 arcminutes in 1 degree, and 60 arcseconds in 1 arcminute — the same base-60 structure as minutes and seconds of time. So 1 degree equals 3,600 arcseconds total.

What angle is one full revolution?+

One revolution (also called a turn or cycle) equals 360 degrees, or 2π radians, or 400 gradians. It represents one complete trip around a circle back to the starting point.

Why does the calculator sometimes show an exact π fraction instead of a decimal?+

When the angle you're converting to radians is a clean fraction of a full circle — like 30°, 45°, 60°, 90°, or 180° — the exact radian value is a simple fraction of π (such as π/6 or π/2). Showing that exact fraction is more precise than rounding to a decimal, since π is irrational and any decimal representation is an approximation.

What does 2π radians represent?+

2π radians is a full circle — the same angle as 360° or 1 revolution. Since a circle's circumference is 2π times its radius, sweeping through 2π radians traces the entire circumference exactly once.

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Angle Converter

Convert an angle

Result

Common angles reference

Exact π form

Exact radians

How to Use This Calculator

Enter the angle value

Pick the from and to units

Read the result

Formula & Methodology

Key Terms Explained

Real-World Examples

Right angle

Full turn

Fine measurement

Understanding the Angle Converter

How the conversion works

Why exact π form matters

When to use each unit

Frequently Asked Questions

Get a branded PDF of your results

Angle Converter

Convert an angle

Result

Common angles reference

Exact π form

Exact radians

How to Use This Calculator

Enter the angle value

Pick the from and to units

Read the result

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Angle Converter

How the conversion works

Why exact π form matters

When to use each unit

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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