How To Convert Angle Units

An angle measures a rotation, and several units describe the same rotation in different ways. The most familiar is the degree, which splits a full turn into 360 equal parts.

A radian is the angle whose arc length equals the radius. Because the circumference of a circle is two pi times the radius, a full turn is two pi radians, which makes one radian about 57.2957795 degrees.

A gradian, also called a gon, splits a right angle into 100 parts, so a full turn is 400 gradians and a right angle is exactly 100 gradians. Surveying and some engineering fields use it because the numbers are decimal.

A minute of arc is one sixtieth of a degree, and a second of arc is one sixtieth of a minute, or one three thousand six hundredth of a degree. These tiny units are used in astronomy, optics and navigation, where a degree is far too coarse.

A turn, or revolution, is simply one full rotation and equals 360 degrees, 400 gradians or two pi radians. It is the natural unit for anything that spins.

The converter routes every value through a single base unit, the degree. Each unit has a fixed size in degrees, and a conversion multiplies the value by the source unit's factor and then divides by the target unit's factor.

Because radians relate to degrees through pi, which is irrational, a radian conversion is only rounded at the point of display. The underlying factor, 180 divided by pi, is exact.

Degrees to radians is the conversion you will use most: multiply the degrees by pi and divide by 180, which is the same as multiplying by 0.0174532925.

Radians to degrees reverses it: multiply the radians by 180 and divide by pi, which is the same as multiplying by 57.2957795.

Gradians sit neatly between the two. One gradian is 0.9 degrees, so a right angle is 100 gradians and a full turn is 400, which is why some survey instruments are graduated in gradians rather than degrees.

Arcminutes and arcseconds subdivide the degree by sixty twice. One degree is sixty arcminutes and one arcminute is sixty arcseconds, so a single degree contains three thousand six hundred arcseconds.

A common mistake is to confuse arcminutes and arcseconds, which measure angles, with minutes and seconds of time. In astronomy the two appear side by side, and a unit slip changes an answer by a factor of fifteen or more.

Engineers and machinists often work in degrees but program in radians, so a conversion sits behind every toolpath and rotation command. Keeping the factor at full precision avoids a slow drift across a long series of moves.

Astronomers describe the apparent size of objects in arcseconds, where a single arcsecond is a tiny slice of the sky. A planet a few arcseconds across looks small in a telescope, but the number is precise enough to track its motion.

Navigation writes angles as degrees, minutes and seconds of arc rather than a decimal. A position of forty degrees, thirty minutes and zero seconds is forty and a half degrees in decimal form.

Because pi is irrational, no radian value is ever exactly a finite decimal. The converter keeps the exact ratio internally and rounds only when it displays the answer, which is the right order of operations.

When several conversions are chained, convert straight to the target unit rather than stepping through intermediate ones. Each intermediate rounding compounds, and a two-step conversion can drift noticeably from a single-step one.

The calculator models the fixed relationships between the units and nothing more. It does not know the precision of your source measurement, so treat the output as exact arithmetic on the number you entered.