Degrees Minutes Seconds Calculator
Convert between DMS, decimal degrees, radians, and gradians. Perform angle format conversions with high precision.
Degrees Minutes Seconds Calculator
Calculator
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Formula: DD = D + M/60 + S/3600 | Radians = DD x PI/180
Worked example — 45 deg 30 min 15.5 sec = 45.504306 degrees = 0.794264 radians
Formula
DD = D + M/60 + S/3600 | Radians = DD x PI/180
Converting DMS to decimal degrees sums the degrees plus minutes/60 plus seconds/3600. Converting to radians multiplies decimal degrees by pi/180. Gradians equal decimal degrees times 10/9. These conversions are reversible and exact.
Worked Examples
Example 1: Navigation Coordinate
Problem:Convert 45 degrees 30 minutes 15.5 seconds to decimal degrees.
Solution:DD = D + M/60 + S/3600 DD = 45 + 30/60 + 15.5/3600 DD = 45 + 0.5 + 0.004306 DD = 45.504306 degrees Radians = 45.504306 x PI/180 = 0.794264 rad
Result:45 deg 30 min 15.5 sec = 45.504306 degrees = 0.794264 radians
Example 2: Decimal to DMS
Problem:Convert 73.985833 decimal degrees to DMS format.
Solution:Degrees = 73 (integer part) Remaining = 0.985833 x 60 = 59.15 minutes Minutes = 59 Seconds = 0.15 x 60 = 9.0 seconds Result: 73 deg 59 min 9.0 sec
Result:73.985833 degrees = 73 deg 59 min 9.0 sec
Frequently Asked Questions
What are degrees, minutes, and seconds in angle measurement?
Degrees, minutes, and seconds (DMS) is a way of expressing angles where one full rotation equals 360 degrees. Each degree is divided into 60 minutes (also called arc-minutes), and each minute is divided into 60 seconds (arc-seconds). This system originated with the ancient Babylonians who used a base-60 number system. DMS is widely used in navigation, astronomy, surveying, and geographic coordinate systems to express latitude and longitude with high precision.
How do I convert DMS to decimal degrees?
To convert DMS to decimal degrees, add the degrees to the minutes divided by 60, plus the seconds divided by 3600. The formula is DD = D + M/60 + S/3600. For negative values (south latitude or west longitude), apply the negative sign to the final result. For example, 45 degrees 30 minutes 15.5 seconds becomes 45 + 30/60 + 15.5/3600 = 45 + 0.5 + 0.004306 = 45.504306 degrees.
What is the difference between radians and degrees?
Degrees divide a full circle into 360 equal parts, while radians use the ratio of arc length to radius. One full rotation equals 2 times pi radians (approximately 6.2832 radians). To convert degrees to radians, multiply by pi/180. Radians are the natural unit for calculus and physics because they simplify many formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. Most programming languages and scientific calculators use radians by default.
What are gradians and when are they used?
Gradians (also called gons or grads) divide a right angle into exactly 100 parts, so a full circle is 400 gradians. This makes them convenient for surveying and civil engineering where right angles and percentages are common. One gradian equals 0.9 degrees. Gradians are standard in many European surveying instruments and are used in slope calculations where the relationship between horizontal and vertical angles is important. Most scientific calculators include a gradian mode alongside degrees and radians.
How precise is one arc-second?
One arc-second of latitude corresponds to approximately 30.87 meters (101.3 feet) on Earth's surface. One arc-second of longitude varies with latitude, equaling about 30.87 meters at the equator and decreasing to zero at the poles. This means GPS coordinates expressed to tenths of arc-seconds are accurate to about 3 meters. For sub-meter precision, coordinates need to be expressed to hundredths of arc-seconds or better, which is why modern GPS uses decimal degrees with many decimal places.
How do you convert decimal degrees back into degrees, minutes, and seconds?
Take the whole number part of the decimal degrees as the degrees. Multiply the remaining fraction by 60 and its whole part becomes the minutes. Multiply whatever is left of that by 60 to get the seconds. For 45.504306 degrees, 0.504306 times 60 gives 30.2584, so 30 minutes, and 0.2584 times 60 gives about 15.5 seconds, recovering 45 degrees 30 minutes 15.5 seconds. Round only at the seconds stage, never the intermediate minutes.
What is the DDM format and why do marine and aviation systems prefer it?
Degrees and decimal minutes keeps whole degrees and expresses the remainder as a decimal minute, such as 45 degrees 30.258 minutes, dropping the seconds field entirely. Charts and aviation navigation databases favour it because one minute of latitude equals one nautical mile, so a decimal minute translates directly into distance along a meridian. Confusing DDM with DMS is a classic position blunder: reading 30.258 as 30 minutes and 258 seconds shifts a fix by kilometres.
How should the sign be handled when converting negative coordinates?
The minus sign belongs to the coordinate as a whole rather than to the individual fields. Convert the absolute values of degrees, minutes, and seconds first, then apply the sign to the finished decimal result, because minutes and seconds are always recorded as positive magnitudes. In hemisphere notation the sign is replaced by a letter, with south latitudes and west longitudes negative. Applying the negative to the minutes term as well is a frequent cause of coordinates landing in the wrong hemisphere.
How large is one arc-minute expressed in radians and gradians?
An arc-minute is one sixtieth of a degree, about 0.0166667 degrees. Multiplying by pi over 180 gives roughly 0.000290888 radians, equivalently pi divided by 10,800. In gradians, where a full circle is 400 gon rather than 360 degrees, an arc-minute equals one sixtieth times ten ninths, or about 0.0185185 gon. These small-angle equivalents matter in optics and astronomy, where instrument resolution is routinely specified in arc-minutes or arc-seconds.
What is the difference between an azimuth and a quadrant bearing in surveying?
An azimuth is a single angle measured clockwise from north through the full 360 degrees, so due west is 270 degrees. A quadrant bearing divides the compass into four 90 degree quadrants and names the direction relative to north or south, such as north 45 degrees 30 minutes east. Azimuths suit coordinate geometry and computation, while quadrant bearings appear on older deeds and property descriptions. Converting between them requires tracking which quadrant the direction falls into.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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