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To convert from degree (deg) to gradian (grad), use the following formula:
gradian (grad)
= 109× degree (deg)
= 1.1111111111111111111× degree (deg)
Let's convert 5 degree (deg) to gradian (grad).
Using the formula:
5 × 1.1111111111111111111 = 5.5555555555555555555
Therefore, 5 degree (deg) is equal to 5.5555555555555555555 gradian (grad).
How many degrees are in one gradian? One gradian (grad) contains 0.9 degrees (deg) — the inverse of the factor above. Multiplying by 1.1111111111111111111 takes you from degrees to gradians; multiplying by 0.9 brings you back.
Put in words: one degree equals 1.1111111111111111111 gradians, so the degree is the larger unit of this pair. Converting between them never changes the amount of angle being measured — only the size of the unit you count it in.
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.
While most of us learn about measuring angles in degrees, there's another unit of angle measurement called the gradian.
A gradian (often abbreviated as grad) is a unit of angle equal to 0.9 (or 9/10) of a degree. It's an alternative to degrees and radians, designed to make some calculations simpler.
Unlike the familiar 360-degree system, a full circle is divided into 400 gradians.
This base-100 system simplifies many geometric calculations:
This makes it an intuitive framework for certain types of angular math.
The conversion between gradians and degrees is straightforward.
Here is the simple gradians to degrees formula:
For example, to convert 200 gradians to degrees, you would calculate: 200 grad × 0.9 = 180 deg.
When comparing radians vs. gradians, the relationship is key for trigonometry and higher math.
A full circle is 400 gradians or 2π radians.
This means that 200 gradians are equal to π radians. The conversion factor, π/200, is fundamental for calculations that bridge these two systems.
So, why use this system at all?
Gradians are primarily used in specific professional fields like surveying and some branches of engineering, especially in Europe.
The decimal-based nature of the 400-gradian circle simplifies calculations involving right angles, making fieldwork and topographical analysis more efficient.
The gradian is the rarest of the three common angle units, but it has not disappeared — it has retreated into a few specific places.
That last point is worth a warning. A calculator left in GRAD mode returns answers that look reasonable but are wrong by a factor of 0.9 against degrees. If a trigonometric result is off by roughly ten percent and nothing else explains it, check the mode before checking the formula.
The quick conversions are:
Here are some quick reference conversions from degree (deg) to gradian (grad):
| degrees | gradians |
|---|---|
| 0.000001 deg | 0.0000011111111111111111111 grad |
| 0.001 deg | 0.0011111111111111111111 grad |
| 0.1 deg | 0.11111111111111111111 grad |
| 1 deg | 1.1111111111111111111 grad |
| 2 deg | 2.2222222222222222222 grad |
| 3 deg | 3.3333333333333333333 grad |
| 4 deg | 4.4444444444444444444 grad |
| 5 deg | 5.5555555555555555555 grad |
| 6 deg | 6.6666666666666666666 grad |
| 7 deg | 7.7777777777777777777 grad |
| 8 deg | 8.8888888888888888888 grad |
| 9 deg | 9.9999999999999999999 grad |
| 10 deg | 11.111111111111111111 grad |
| 20 deg | 22.222222222222222222 grad |
| 30 deg | 33.333333333333333333 grad |
| 40 deg | 44.444444444444444444 grad |
| 50 deg | 55.555555555555555555 grad |
| 100 deg | 111.11111111111111111 grad |
| 1000 deg | 1111.1111111111111111 grad |
| 10000 deg | 11111.111111111111111 grad |