Why does a circle have 360 degrees?

Historically from Babylonian astronomy, around 2000 BC. The Babylonians used a base-60 (sexagesimal) system and observed roughly 360 days per year. The number 360 is highly divisible: it splits evenly into 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360, ideal for geometric constructions without a calculator. Degrees have survived from antiquity into today.

One radian is the angle at which the arc length equals the radius. A full circle has circumference 2πr, so 2π radians. That makes radians "natural" for maths: sin′(x) = cos(x) only when x is in radians; otherwise you get a 180/π factor. Programming languages like JavaScript’s Math.sin() expect radians. π ≈ 3.14159, so 180° = π rad ≈ 3.14159 rad.

Who uses gradians (gon)?

Gon (gradians) is mainly used in surveying and the mineral industry, especially in Central Europe (Germany, France, Switzerland). A full circle is 400 gon, so a right angle is 100 gon. The advantage is easier decimal maths: 45° = 50 gon, 30° = 33.33 gon. Calculators often have a "DEG / RAD / GRAD" mode, where GRAD means gon.

What are arc-minutes and arc-seconds?

Subdivisions of a degree: 1° = 60′ (arc-minutes), 1′ = 60″ (arc-seconds). Used in GPS coordinates, astronomy, sextant navigation and optics. On Earth, one arc-minute along the meridian is one nautical mile (1852 m). One arc-second along the meridian is about 30.9 m. Astronomical observations often measure in milli-arcseconds (mas) and micro-arcseconds (μas) for the highest precision.

Angle Converter - Degrees, Radians, Gradians, Turns

Degrees	rad	gon	turns
90°	π/2 ≈ 1.5708	100	0.25
180°	π ≈ 3.1416	200	0.5
270°	3π/2 ≈ 4.7124	300	0.75
360°	2π ≈ 6.2832	400	1

Degrees

rad

gon

turns

90°

π/2 ≈ 1.5708

100

0.25

180°

π ≈ 3.1416

200

0.5

270°

3π/2 ≈ 4.7124

300

0.75

360°

2π ≈ 6.2832

400

How angle conversion is calculated

Angles are converted via degrees as the base unit. Radians express the ratio of arc length to radius, while gradians (gon) divide a right angle into 100 parts.

1 radian = 180 / π ≈ 57.29578 degrees
1 gradian (gon) = 0.9 degrees
1 turn = 360 degrees

About this tool

The angle converter maps between degrees, radians, gradians (gon), arc-minutes, arc-seconds and turns (full rotations). Useful for programming (JavaScript math uses radians), surveying and GPS (degrees, minutes, seconds), maths work, navigation, astronomy and CAD. All in the browser, updated in real time.

How to use it

Type a value in any field.
See all other units update in real time.
Switch to the DMS tab for degrees-minutes-seconds (like GPS coordinates).
Copy the result with one click.

Examples

Right angle

Input90°

Outputπ/2 rad = 1.5708 rad = 100 gon

GPS coordinate

Input59° 54′ 40″ N

Output59.9111° N

Degrees-minutes-seconds (DMS) is the standard for GPS and navigation. One degree = 60 minutes, one minute = 60 seconds. Decimal degrees are easier for databases and APIs.

Full rotation

Input1 full turn

Output360° = 2π rad = 400 gon

Common use cases

JavaScript programming: convert degrees to radians for Math.sin/cos.
Surveying and GPS: DMS coordinates to decimal degrees.
Maths problems: exact values with π (π/6, π/4, π/3, π/2).
Navigation: compass headings in degrees and radians.
CAD and 3D modelling: rotation angles as radians or degrees.

Frequently asked questions

Why does a circle have 360 degrees?: Historically from Babylonian astronomy, around 2000 BC. The Babylonians used a base-60 (sexagesimal) system and observed roughly 360 days per year. The number 360 is highly divisible: it splits evenly into 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360, ideal for geometric constructions without a calculator. Degrees have survived from antiquity into today.
What are radians?: One radian is the angle at which the arc length equals the radius. A full circle has circumference 2πr, so 2π radians. That makes radians "natural" for maths: sin′(x) = cos(x) only when x is in radians; otherwise you get a 180/π factor. Programming languages like JavaScript’s Math.sin() expect radians. π ≈ 3.14159, so 180° = π rad ≈ 3.14159 rad.
Who uses gradians (gon)?: Gon (gradians) is mainly used in surveying and the mineral industry, especially in Central Europe (Germany, France, Switzerland). A full circle is 400 gon, so a right angle is 100 gon. The advantage is easier decimal maths: 45° = 50 gon, 30° = 33.33 gon. Calculators often have a "DEG / RAD / GRAD" mode, where GRAD means gon.
What are arc-minutes and arc-seconds?: Subdivisions of a degree: 1° = 60′ (arc-minutes), 1′ = 60″ (arc-seconds). Used in GPS coordinates, astronomy, sextant navigation and optics. On Earth, one arc-minute along the meridian is one nautical mile (1852 m). One arc-second along the meridian is about 30.9 m. Astronomical observations often measure in milli-arcseconds (mas) and micro-arcseconds (μas) for the highest precision.

Technical background

All angles are converted via radians as the mathematically canonical base. Constants: 1° = π/180 rad ≈ 0.01745 rad, 1 rad = 180/π ° ≈ 57.2958°, 1 gon = π/200 rad = 0.9° ≈ 0.01571 rad, 1 turn = 2π rad = 360° = 400 gon, 1′ (arc-minute) = 1/60° = π/10800 rad, 1″ (arc-second) = 1/3600° = π/648000 rad ≈ 4.848·10⁻⁶ rad. π is an irrational constant, approximately 3.14159265358979323846... JavaScript’s Math.PI gives about 15 decimals of precision. Conversion formulas: rad = deg·π/180, deg = rad·180/π, gon = deg·10/9, deg = gon·9/10. For DMS ↔ decimal: dd = d + m/60 + s/3600 (and back: d = floor(dd), m = floor((dd−d)·60), s = ((dd−d)·60 − m)·60). The tool uses JavaScript Number (IEEE 754 double), giving 15–17 significant decimal digits, far beyond practical need.

Angle reference table