One degree is 1.1111111 gradians, because a full circle is 400 gradians instead of 360. The gradian was the metric system's attempt to rationalise angle measurement, and it survives in exactly one profession.
About the units
The gradian, also called the gon or the grade, divides the right angle into 100 parts rather than 90, giving 400 to the full circle. It was introduced during the French metric reforms of the 1790s alongside decimal time and the decimal calendar, and unlike those two it was never quite abandoned. The appeal is that a quarter turn is a round 100, so bearings and slopes come out in decimal fractions without the sexagesimal minutes and seconds that degrees drag along. Surveying in France, Germany and parts of Scandinavia adopted it and has kept it.
The exact factor
The factor is exactly 400/360 = 10/9 = 1.1111... gradians per degree. Useful equivalents: 90° is 100 gon, 45° is 50 gon, 180° is 200 gon, and 1 gon is 0.9°. In radians a gradian is π/200. The decimal structure is the whole point — a bearing of 137.4562 gon needs no conversion to sub-units, where the same bearing in degrees would be written 123° 42′ 38″ and require sexagesimal arithmetic to add or subtract.
Where you meet this conversion
Surveying and civil engineering in continental Europe. Total stations and theodolites sold in France, Germany, Austria, Switzerland and the Nordic countries can be set to gon, and cadastral records in those countries use it. Road and railway alignment calculations are done in gon where the tradition holds. Scientific calculators have a GRAD mode alongside DEG and RAD, and accidentally leaving a calculator in it is a classic source of wrong answers in exams — the results are plausible but consistently off by eleven per cent.