Angle Converter Online — Convert Degrees, Radians, Gradians & More

Instantly convert between angle units: degrees, radians, gradians, turns, arcminutes, arcseconds, and milliradians

Angle Conversion
7 units
From
To
Ratio: 360° = 2π rad = 400 grad = 1 turn
Result

Enter a value to convert

Common Reference Angles
30° — π/6 rad · sin 30° = 0.5

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45° — π/4 rad · sin 45° = cos 45° = √2/2

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90° — π/2 rad · right angle

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180° — π rad · straight angle

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270° — 3π/2 rad · three-quarter turn

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360° — 2π rad · full rotation

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Angle Measurement Units
Degrees (°)

The most common angle unit. A full circle = 360°. One degree = 60 arcminutes = 3,600 arcseconds. Used in geometry, navigation, geography, and everyday life

Radians (rad)

The standard unit in mathematics and programming. One radian is the angle where the arc length equals the radius. Full circle = 2π ≈ 6.2832 rad. Formula: rad = deg × π / 180

Gradians (gon)

Divide a circle into 400 parts. A right angle = 100 grad. Used in surveying and some European engineering traditions. Formula: grad = deg × 10 / 9

Turns (rev)

One full rotation = 1 turn = 360° = 2π rad. Convenient for describing rotational motion, cycles, and frequencies. Half turn = 0.5 turn = 180°

Arcminutes (′)

One-sixtieth of a degree. 1° = 60′. Used in astronomy, navigation, and cartography for precise positioning. Denoted with a prime symbol (′)

Arcseconds (″)

One-sixtieth of an arcminute. 1′ = 60″, so 1° = 3,600″. The smallest standard angular unit. Used in astronomy to measure small angular distances between stars

Milliradians (mrad)

One-thousandth of a radian. 1 rad = 1,000 mrad. Widely used in ballistics, optics, and military applications. At 1 km distance, 1 mrad corresponds to ~1 meter of displacement

Conversion Formulas
Degrees → Radians

rad = deg × π / 180

Radians → Degrees

deg = rad × 180 / π

Degrees → Gradians

grad = deg × 10 / 9

Degrees → Turns

turn = deg / 360

Degrees → Arcminutes

arcmin = deg × 60

Degrees → Arcseconds

arcsec = deg × 3,600

Degrees → Milliradians

mrad = deg × 1,000π / 180

Frequently Asked Questions
How do you convert degrees to radians?

Multiply the degree value by π / 180 (≈ 0.01745). For example: 90° × π / 180 = π/2 ≈ 1.5708 rad. This is the most common angle conversion, required for trigonometric functions in most programming languages and scientific calculators.

How do you convert radians to degrees?

Multiply the radian value by 180 / π (≈ 57.2958). For example: π rad × 180 / π = 180°. The reverse formula: degrees = radians × 57.29577951. Most calculators have a dedicated button for this conversion.

How many radians are in a full circle?

A full circle = 2π ≈ 6.2832 radians = 360 degrees = 400 gradians = 1 turn. A half circle = π ≈ 3.1416 rad = 180°. A quarter circle (right angle) = π/2 ≈ 1.5708 rad = 90°.

What is the difference between gradians and degrees?

Degrees divide a circle into 360 parts, gradians divide it into 400. A right angle = 90° = 100 grad. Gradians are convenient for decimal calculations in surveying and land measurement. Formula: gradians = degrees × 10/9.

What are arcminutes and arcseconds?

An arcminute (′) = 1/60 of a degree. An arcsecond (″) = 1/60 of an arcminute = 1/3,600 of a degree. They are used for precise measurements in navigation, astronomy, and cartography. For example, GPS coordinates: 40°44′54″N, 73°59′08″W (New York City).

What are milliradians used for?

Milliradians (mrad) are used in ballistics, military applications, and optical sights. The key advantage: at a distance of 1,000 m, one milliradian corresponds to approximately 1 meter of displacement. This simplifies aiming and fire correction. A full circle ≈ 6,283 milliradians.

Why do programming languages use radians instead of degrees?

Trigonometric functions (sin, cos, tan) in JavaScript, Python, C++, Java, and most other languages accept arguments in radians. This is the mathematical standard that simplifies derivative and integral calculations. Before using Math.sin(), convert degrees: x = degrees * Math.PI / 180.

How do you convert DMS (degrees-minutes-seconds) to decimal degrees?

Formula: decimal degrees = degrees + minutes/60 + seconds/3,600. For example: 40°44′54″ = 40 + 44/60 + 54/3,600 = 40.7483°. Reverse: multiply the fractional part by 60 to get minutes, then the fractional part of minutes by 60 to get seconds.

Angle Converter — How to Convert Between Angle Units

Angle measurement units are used across mathematics, physics, engineering, navigation, astronomy, and programming. The most common are degrees and radians, but depending on the field, you may also encounter gradians, arcminutes, arcseconds, and milliradians. This angle converter lets you instantly convert any value between all 7 units.

Degrees to Radians — The Most Common Conversion

Converting degrees to radians and back is the most frequently needed angle conversion. In everyday life we use degrees (360° = full circle), but in mathematics and programming the standard is radians (2π = full circle). The formula is straightforward: radians = degrees × π / 180. For example, a right angle of 90° = π/2 ≈ 1.5708 radians.

Gradians — The Surveying System

Gradians (also called gons or grads) divide a full circle into 400 equal parts instead of 360. This makes a right angle exactly 100 gradians, which is convenient for decimal-based calculations in land surveying and topography. Gradians are primarily used in continental Europe for surveying work.

Arcminutes and Arcseconds — Precision Positioning

The DMS system (degrees-minutes-seconds) is widely used in navigation, cartography, and GPS coordinates. One degree equals 60 arcminutes and 3,600 arcseconds. At the Earth's surface, one arcsecond corresponds to approximately 31 meters, making it precise enough for most geographic applications.

Milliradians in Military and Optics

A milliradian (mrad) is one-thousandth of a radian. Its key property is the linear relationship at long distances: 1 mrad at 1,000 meters equals approximately 1 meter of displacement. This makes milliradians essential for rifle scopes, artillery aiming, and optical instrument calibration. A full circle contains approximately 6,283 milliradians.

Radians in Programming

Most programming languages — including JavaScript, Python, C++, Java, and Swift — use radians as the default unit for trigonometric functions. The functions Math.sin(), Math.cos(), and Math.tan() all expect their arguments in radians. To use degrees, first convert: radians = degrees * Math.PI / 180. Some languages offer helper functions like Python's math.radians() and math.degrees().

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