Angle units:
moving between degrees, radians and arcseconds

Why angles have more than one unit

A full circle can be divided in several ways, and each choice stuck for a reason. Degrees are traditionally traced to Babylonian astronomy, and 360 divides evenly by so many numbers that mental fractions are easy. Radians come from geometry: a radian is the angle subtended by an arc whose length equals the radius, which makes calculus formulae cleaner.

Arcminutes and arcseconds are the fine print of degrees. Astronomers, surveyors and navigators need angles much smaller than a degree, so each degree is cut into 60 arcminutes and each arcminute into 60 arcseconds. The same units give latitude and longitude on a map their degrees-minutes-seconds notation.

There is one more unit, the gradian, which splits a right angle into 100 parts so that a full circle has 400. It appears on some surveying instruments and calculators, but the other three do nearly all the work.

The two anchor facts and the formulas

The multiplier π ÷ 180 is 0.0174533, so degrees to radians is a single multiplication. Going the other way uses 57.2958. Arcminutes to degrees divides by 60, and arcseconds to degrees divides by 3,600. To go from arcseconds to arcminutes, divide by 60 once.

For quick mental work, treat 1 radian as a little under 60 degrees and 1 degree as a little under 0.02 radian. A right angle is then about 1.57 radians, a straight line about 3.14 and a full turn about 6.28. If an answer for a right angle comes out near 90 or near 0.5, it is probably in the wrong unit.

Common angles at a glance

Notice the exact column. In maths and physics you usually keep a multiple of π and only convert to decimals at the end, which avoids rounding errors in the middle of a calculation.

Memorise the 30°, 45°, 60° and 90° rows and you can derive most others by adding or doubling. For example, 120° is twice 60°, so 2π/3, and 135° is 90° plus 45°, so 3π/4.

Worked examples: Earth's tilt and a latitude

Going the other way: Hyderabad's latitude is about 17.385° north. The whole degrees are 17. The remainder 0.385 × 60 = 23.1 arcminutes gives 23′, and the leftover 0.1 × 60 = 6 arcseconds, so 17° 23′ 6″ N.

One more: 37.5° in radians is 37.5 × 0.0174533 = 0.6545. A calculator shows more digits, but four decimal places are plenty for most work.

Where radians earn their keep: arc length and rotation

Radians look odd until you use them to measure arcs. When an angle is in radians, the length of the arc it cuts from a circle is simply the radius multiplied by the angle. Degrees need an extra factor of π ÷ 180 every time, which is why physics and engineering formulas assume radians.

A wheel of radius 0.3 m that turns through 2 radians moves a point on its rim by 0.6 m. Rotation speeds follow the same idea: one revolution per minute is 2π ÷ 60 = 0.1047 radians per second.

The same relationship explains latitude. With the Earth's radius at about 6,371 km, one degree of latitude spans 6,371 × 0.0174533 = 111.2 km along a meridian. One arcminute is a sixtieth of that, about 1.85 km, which is the origin of the nautical mile, and one arcsecond is about 31 m.

How small is an arcsecond?

An arcsecond is 1/3,600 of a degree, which is 4.848 millionths of a radian. At a distance of 1 kilometre, that angle spans about 4.85 mm; a 2 cm coin viewed from about 4 km away covers roughly one arcsecond. The Moon, which looks small, spans about half a degree, or roughly 1,800 arcseconds.

Because the angles are so small, astronomers and surveyors use the small-angle idea: for tiny angles in radians, the angle approximately equals the arc length divided by the distance. The conversion to radians is the step that lets you use that shortcut with metric lengths.

Surveyors and astronomers also meet milliarcseconds, a thousandth of an arcsecond, when measuring the positions of stars. Those numbers are tiny: a single milliarcsecond is 4.8 × 10⁻⁹ radians, and getting a result at that scale needs the full precision of π rather than a rounded 3.14.

Mistakes to watch for

• Calculator mode. sin(30) with the calculator in radian mode gives −0.988, not 0.5, because it reads 30 as 30 radians. Check the DEG/RAD indicator before every trig problem.
• Dividing by 100 instead of 60 for minutes. Decimal fractions of a degree are not the same as minutes: 0.5° is 30′, and 0.30° is 18′.
• Writing 23.26.21 and reading it as 23.26° rather than 23° 26′ 21″.
• Mixing up the symbol for minutes of arc (′) with minutes of time. In astronomy, right ascension uses hours, minutes and seconds of time, where 1 hour = 15°.
• Rounding π to 3.14 early. For anything beyond a rough check use the calculator's π key or at least 3.14159.

A good habit is to sanity-check each answer by size. A degrees-to-radians result should be about one sixtieth of the degree figure (0.01745 times it). An arcsecond-to-degree result should be a very small number, and an arcminute-to-degree result slightly bigger. If the answer is off by an order of magnitude, a factor of 60 or 3,600 was applied the wrong way round.