How to use the Angle Converter
- Enter the angle value.
- Select the unit that value is expressed in.
- Read the equivalent in degrees, radians and gradians.
- Use the DMS row when working with survey bearings or astronomical coordinates.
- Copy the radian value when feeding an angle into programming or spreadsheet trigonometry.
How the calculation works
Four units divide the circle in four different ways. Degrees split it into 360, inherited from Babylonian base-60 astronomy and convenient because 360 has so many whole divisors. Radians split it into 2π, the only unit in which arc length equals radius times angle without a conversion factor. Gradians split it into 400, a metric-era invention that makes a right angle exactly 100. Turns simply count full revolutions.
Degrees-minutes-seconds subdivides each degree into 60 minutes and each minute into 60 seconds, the sexagesimal convention still used in surveying, navigation and astronomy. One second of arc is roughly 31 metres of latitude on the Earth's surface, which is why land boundaries and star positions are quoted to that precision and why decimal-degree rounding can shift a boundary line by metres.
The unit that matters most in practice is the radian, because every programming language, spreadsheet and scientific library expects trigonometric arguments in radians. Passing degrees to a sine function is the single most common numerical bug in engineering code: it produces plausible-looking numbers that are silently, badly wrong. Convert first, always.
radians = degrees × π/180; gradians = degrees × 10/9; turns = degrees/360; DMS: d° m′ s″ with m = frac(deg)×60Source: SI-accepted angle units per BIPM SI Brochure, 9th edition, Table 8; DMS per IAU coordinate conventions.
Worked example
A survey bearing is recorded as 47.3625°. Express it for a field book and for a spreadsheet.
- Whole degrees: 47.
- Minutes: 0.3625 × 60 = 21.75, so 21 whole minutes.
- Seconds: 0.75 × 60 = 45.
- Radians for the spreadsheet: 47.3625 × π/180 = 0.82663.
47° 21′ 45″ for the field book, 0.82663 radians for any SIN or COS formula, and 52.625 gradians on European instruments.
Frequently asked questions
Why do programming languages use radians?+
Because the calculus of trigonometric functions is clean only in radians — the derivative of sine is cosine exactly, with no stray conversion constant.
Who still uses gradians?+
Some European surveying instruments and older French engineering practice. A right angle being exactly 100 grad simplifies quadrant arithmetic.
How precise is one arcsecond?+
About 31 metres of latitude at the Earth's surface, or the width of a coin seen from four kilometres away.
What is a mil?+
A military unit, usually 6,400 to a full circle, chosen because one mil subtends roughly one metre at a kilometre — handy for artillery correction.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.