Convert angle

About the units

• Degrees (°) – A degree is one 360th of a turn, π/180 radians. The degree is not an SI unit, but it is one of the long-standing units of angle in the SI Brochure, and it is a legal unit throughout the EU. It is used for maps, coordinates, building angles and compass bearings. Coordinates can be written in decimal degrees, in degrees with decimal minutes or in degrees, minutes and seconds: 15.51° is 15°30.6′, that is 15°30′36″. And there is the trap: 15.51° is not 15°51′.
• Radians (rad) – The radian is the SI coherent unit of plane angle and a legal unit throughout the EU. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius; a right angle is π/2 radians. That makes one radian 180/π degrees, 57.29578°, and a full turn 2π radians. All mathematics and physics count in radians, and angular velocity is given in radians per second. The trap is the calculator mode: sin(30) is 0.5 in degrees but −0.988 in radians.
• Milliradians (mrad) – A milliradian is a thousandth of a radian, and this conversion uses exactly that thousandth: a full turn is 2π radians, so 6,283.19 milliradians. The unit is handy because angle and distance line up directly – one milliradian covers 10 centimetres at 100 metres. Riflescopes with milliradian turrets move the point of impact 1 centimetre at 100 metres per click of 0.1 milliradian. The trap is the military mil, which the US Army map-reading manual defines as one 6,400th of a turn, 1.8% smaller than a milliradian.
• Gradians (gon) – A gradian, or gon, is one 400th of a turn and therefore exactly 0.9 degrees – 100 gon make a right angle. The unit is listed among the legal units in the EU units directive, which allows prefixes on gon but not on the degree. It lives on in surveying and geodesy: the Swedish national grid is called RT 90 2.5 gon V, its central meridian lying 2.5 gon, 2.25 degrees, west of the reference meridian. The GRA mode on a calculator is gon, not degrees.
• Arcminutes (′) – An arcminute is one sixtieth of a degree, π/10,800 radians. It belongs to the long-standing units of angle in the SI Brochure and is a legal unit in the EU. Arcminutes are used in navigation, astronomy, geodesy and optometry, and geographical coordinates are often written as degrees with decimal minutes. The traps are the symbols: ′ is an arcminute but also a foot in American writing, and min is a minute of time. N 59° 19.8′ is not 59.198°.
• Arcseconds (″) – An arcsecond is one sixtieth of an arcminute, one 3,600th of a degree or π/648,000 radians. It belongs to the long-standing units of angle in the SI Brochure and is a legal unit in the EU. The arcsecond is the everyday measure of astronomy, and the SI Brochure also mentions milli- and microarcseconds. The International Astronomical Union defines the parsec from it: the distance at which one astronomical unit subtends one arcsecond, exactly 648,000/π astronomical units. The trap is the sign ″, which also means inch.
• Turns (rev) – A turn is one full rotation: 360 degrees, 2π radians or 400 gon. The EU units directive lists the revolution as a legal unit, with two caveats – it is not in the General Conference’s lists, and no international symbol exists, so abbreviations such as r and rev are practice rather than a standard. The SI Brochure notes that “revolution” is not an SI unit and warns of the factor 2π: one turn per second is a frequency of one hertz but an angular velocity of 2π radians per second.