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Multiple conversions
To convert from degree (deg) to radian (rad), use the following formula:
radian (rad)
= π180× degree (deg)
= 0.017453292519943294444× degree (deg)
To convert from degree (deg) to gradian (grad), use the following formula:
gradian (grad)
= 109× degree (deg)
= 1.1111111111111111111× degree (deg)
To convert from degree (deg) to arcminute (arcmin), use the following formula:
arcminute (arcmin)
= 60× degree (deg)
To convert from degree (deg) to arcsecond (arcsec), use the following formula:
arcsecond (arcsec)
= 60 × 60× degree (deg)
= 3600× degree (deg)
In geometry and everyday life, we measure angles using degrees.
A degree (represented by the universal symbol °) is the basic unit for measuring rotation. It helps us describe the amount of turn between two lines that meet at a point.
For instance, a perfect corner, like the edge of a book, is a 90° right angle, and one complete, full-circle rotation measures 360°.
The standard of a circle containing 360 degrees (360°) originated with the ancient Babylonians.
They used a base-60 (sexagesimal) number system, and 360 was a perfect number for them because it is highly divisible. Its large number of factors made it incredibly easy to perform fractional calculations in fields like astronomy and geometry.
For fields that require extreme precision, like astronomy, cartography (map-making), and GPS navigation, a single degree is broken down into even smaller units:
This system ensures that even the tiniest angles can be measured with high accuracy.
There is no physical reason a full turn must be 360 of anything. The number is an inheritance from Babylonian astronomy, and it survived for two reasons that still matter.
The first is the calendar. Babylonian astronomers worked with a year of roughly 360 days, so the Sun appeared to advance about one degree along the ecliptic each day — a convenient bookkeeping unit for tracking the sky.
The second is arithmetic, and it is the reason the convention outlived the astronomy. 360 has 24 divisors. It splits evenly into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths, twelfths, fifteenths, eighteenths, twentieths, twenty-fourths and more. Every common fraction of a turn lands on a whole number of degrees: a right angle is 90°, a third of a turn is 120°, an eighth is 45°. A base-10 alternative such as the gradian, with 400 to a circle, is better for decimal arithmetic but produces fractions for thirds and sixths — which is roughly why it never displaced the degree outside surveying.
Degrees are for people; radians are for mathematics and for machines.
The distinction is not stylistic. In calculus the derivative of sin(x) is cos(x) only when x is in radians; in degrees an awkward factor of π/180 appears and never leaves. Because of this, essentially every programming language's trigonometric library — JavaScript, Python, C, Java — takes radians, not degrees.
That mismatch is a reliable source of bugs. A value that looks plausible but is wrong by a factor of about 57.3 usually means degrees were passed where radians were expected. The conversions worth memorising are:
Scientific calculators expose the same trap through their DEG / RAD / GRAD mode switch, where an unnoticed mode is the classic cause of an answer that is confidently and completely wrong.