How to Convert Decimal Degrees to Degrees, Minutes, and Seconds

The sexagesimal system, not to be confused with the hexadecimal system, is a number system in which each unit is divided into 60 units belonging to the lower order . There are several physical magnitudes that are represented in this type of systems. One of them is the measurement of the amplitude of an angle, whose main unit of measurement is the degree, which in turn is divided into minutes and seconds following a sexagesimal system.

Probably due to the fact that early clocks marked time in the form of an angle, we also often express time in a similar system in which the primary unit is the hour. As we well know, the hour is divided into 60 minutes and each minute into 60 seconds, so it also represents an example of the use of the sexagesimal system. Two other common examples are geographic coordinates as a function of latitude and longitude.

This type of system can be very convenient for certain applications, but the use of these quantities makes it considerably difficult to carry out such simple mathematical operations as addition, subtraction, multiplication and division. Likewise, when we carry out calculations of magnitudes such as angles or times, it is common for us to express these magnitudes, as well as the results, in the traditional decimal system, which sometimes makes their daily interpretation difficult.

For example, saying that it took us 3,127 hours to get from point A to point B is not understood with the same clarity as if we had said that it took us 3 hours, 7 minutes and 37 seconds. For this reason, it is very important to know how to convert decimal degrees to the sexagesimal system of degrees (°), minutes (‘) and seconds (“).

Converting Decimal Degrees to Degrees, Minutes, and Seconds

Converting from decimal degrees to sexagesimal degrees is not like other unit conversions where you just need to apply a formula and voila! Rather, the procedure is actually a very simple three-step algorithm. We will illustrate these steps using the conversion of the angle 123.456° to degrees, minutes and seconds as an example.

Step 1: Separate the integer part of the number from the decimal part

When we express an angle in decimal degrees, the whole part of the number corresponds to the number of whole degrees, while the decimal part is the one that contains the minor subunits corresponding to minutes and seconds.

In our example, the degrees of the angle in the sexagesimal system will be 123° , while the decimal part, those 0.456° , are what we must now convert to minutes and seconds.

Step 2: Multiply the decimal part by 60 to get the minutes

The next step is to extract the number of minutes from the decimal part. To do this, simply multiply the original decimal part by 60 and then separate the integer part of the result from the new decimal part. The integer part of the result corresponds to the number of minutes in the angle, while the decimal part contains the seconds and must be converted later.

In our example, we multiply

In this case, the integer part 27 corresponds to minutes, while the decimal part, 0.36, which is now in minutes, must be converted to seconds.

Step 3: Multiply the new decimal part by 60 to get the seconds

The last step of the algorithm consists of transforming the decimal part of the minutes to seconds. Again, this is done by multiplying this decimal part by 60 and the result of this multiplication gives the seconds. Normally, the seconds are not divided into smaller units in the sexagesimal system, so the result is expressed in decimal form, if it has them.

In our example, the decimal part of the minutes is 0.36, so the seconds will be:

Finally, the result is expressed by reporting the minutes, degrees and seconds, one after the other followed by the symbols °,’, and ”, respectively. That is to say:

reverse conversion

The procedure to carry out the inverse conversion, that is, to take a number expressed in the sexagesimal system to the decimal system, consists of dividing the minutes by 60, the seconds by 3600 and then adding these two results and the number of degrees.

For example, if we want to convert the latitude of the center of Tokyo, Japan, which is 35°41’22.2” to decimal degrees, the result will be:

References

• Geodata. (nd). Geographic coordinates of Tokyo – Latitude and longitude . Retrieved from https://www.geodatos.net/coordenadas/japon/tokyo
• Ortiz, M. (2014, January 17). Formula to convert decimal degrees to degrees, minutes and seconds . Recovered from https://exceltotal.com/formula-para-convertir-grados-decimales-grados-minutos-y-segundos/
• planetcalc. (2019). Online calculator: Conversion of degrees-minutes-seconds to decimal degrees and vice versa . Recovered from https://es.planetcalc.com/1129/
• AllThe.Info. (nd). Convert Decimal Degrees to Degrees Minutes and Seconds OnLine . Retrieved from https://grados-decimales-a-grados-minutos-y-segundos.todala.info/