Angle Converter
360 degrees is Babylonian; radians are natural.
Convert Angle. 360 degrees is Babylonian; radians are natural. Shows the working, not just the answer.
Converted
90 degrees
1.5708 radians · 100 gradians
Degrees are a Babylonian inheritance, chosen because 360 divides evenly by a great many numbers. Radians are the mathematically natural unit — the angle subtending an arc equal to the radius — which is why calculus formulas are clean in radians and carry conversion factors in degrees.
How the Angle Converter works
Angles across degrees, radians, gradians, turns and the arc units. Radians are the mathematically natural unit, which is why calculus formulas are clean in radians and carry conversion factors in degrees.
Also known as: degrees to radians converter · radians to degrees · gradians to degrees · arcminutes and arcseconds converter
Why 360, and why radians instead
Three hundred and sixty degrees is a Babylonian inheritance from a sexagesimal number system, and a convenient one — 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20 and more. It may also reflect an approximate year in days.
A radian is the angle subtending an arc equal in length to the radius, so a full circle is 2π radians. It is defined by the geometry rather than chosen, which is why it is called the natural unit.
The payoff appears in calculus. The derivative of sine is cosine only when the angle is in radians; in degrees an extra factor of π/180 appears in every formula. Arc length is simply radius times angle, with no conversion factor at all.
The small-angle units
Arcminutes and arcseconds are sixtieths and thirty-six-hundredths of a degree, inherited from the same sexagesimal system that gives 60 minutes to an hour. One arcsecond is roughly the width of a coin seen from four kilometres.
They persist in astronomy and navigation, where angles that small are routine and where degrees would need many decimal places. Stellar parallax is measured in arcseconds, which is where the parsec gets its name.
The NATO mil is a different small unit: 1/6400 of a circle, chosen so that at a thousand metres one mil subtends about one metre. It makes range and correction arithmetic trivial in the field, which is the entire point.
Angles on a curved surface
A triangle's angles sum to 180 degrees on a flat plane. On a sphere they sum to more, and the excess is proportional to the triangle's area — a triangle covering an eighth of a sphere has three right angles.
That makes curvature detectable from inside the surface, without any reference to a surrounding space. It is the basis of differential geometry and, ultimately, of how general relativity describes gravity.
It also matters practically in surveying and navigation over long distances, where treating the Earth as flat accumulates error. Great circle routes are shortest precisely because the surface is curved.
Where to go next
The Angle Converter question rarely arrives on its own. These are the ones that usually come with it:
- Slope Intercept Calculator — A vertical line has undefined slope, not infinite.
- Polygon Area Calculator — Exterior angles always sum to 360°, whatever the shape.
- Circle Area Calculator — A circle fills 78.54% of its square, always.
- Ring Size Calculator — Finger measurement to US, UK, EU and Japanese sizes.
Frequently asked questions
How do I convert degrees to radians?
Multiply by π ÷ 180. Ninety degrees is π/2 radians, which is about 1.5708.
Why 360 degrees in a circle?
A Babylonian inheritance, and a convenient one — 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many more. It may also relate to their approximation of a year in days.
What is a radian?
The angle subtending an arc equal in length to the radius. There are 2π of them in a full circle, which makes arc length simply radius times angle with no conversion factor.
Why do calculus formulas need radians?
Because the derivative of sine is cosine only when the angle is in radians. In degrees an extra factor of π/180 appears everywhere, which is why radians are called the natural unit.
What is a gradian?
One hundredth of a right angle, so 400 to a full circle. It was a metric-era attempt to decimalise angle measurement and it survives mainly in surveying.
What are arcminutes and arcseconds?
Sixtieths and thirty-six-hundredths of a degree, used in astronomy and navigation where very small angles matter. One arcsecond is roughly the width of a coin seen from four kilometres.
What is a solid angle?
The three-dimensional analogue, measured in steradians. A full sphere is 4π steradians, exactly as a full circle is 2π radians.
How do surveyors measure angles?
In degrees, minutes and seconds, or in gradians in some countries. Bearings are measured clockwise from north, which is the opposite convention to mathematical angles.
What is a mil in military use?
A NATO mil is 1/6400 of a circle, chosen because at a thousand metres one mil subtends roughly one metre. That makes range estimation arithmetic straightforward.
Why are there 60 minutes in a degree?
The same Babylonian sexagesimal system that gives 60 minutes to an hour. Degrees, minutes and seconds of arc are a direct inheritance from it.
What is the angle sum of a triangle?
180 degrees on a flat plane. On a sphere it exceeds 180, and the excess is proportional to the triangle's area — which is how curvature is detected without leaving the surface.
How do I convert a gradient to an angle?
Take the arctangent. A 1 in 10 gradient is arctan(0.1), which is about 5.7 degrees — noticeably less than most people estimate.
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