Calculating...
Enter two DMS angles and choose add or subtract to see the result with carry/borrow logic.
What Is DMS (Degrees, Minutes, Seconds)?
Degrees-minutes-seconds, or DMS, is the traditional way to express angles. A degree is split into 60 minutes, and each minute is split into 60 seconds. This sexagesimal system is still the standard for land surveying, navigation, astronomy, and old property deeds because it is easier to read in the field than long decimal strings.
Adding Two DMS Angles Visually
45°30′15″ + 12°45′50″ = 58°16′5″
Carry Breakdown
+50″
Seconds column
65″ = 1′ 5″ · carry 1 minute
+45′
+1′
Minutes column
76′ = 1° 16′ · carry 1 degree
+12°
+1°
Degrees column
45° + 12° + 1° = 58°
Visual example: 45°30′15″ + 12°45′50″. The carry moves from seconds → minutes → degrees.
How to Add DMS Angles by Hand
Start from the smallest unit — seconds. Add the seconds together. If the total reaches 60, carry 1 minute and keep the remainder as seconds. Repeat the same process for minutes: if the total reaches 60, carry 1 degree. Finally add the degrees, including any carried values. For example, 45°30′15″ + 12°45′50″ gives 65 seconds, which becomes 1 minute and 5 seconds. The minutes then become 76, which becomes 1 degree and 16 minutes, giving a final answer of 58°16′5″.
How to Subtract DMS Angles With Borrowing
Subtract seconds first. If the second angle has larger seconds, borrow 1 minute from the first angle, convert it to 60 seconds, and then subtract. Do the same for minutes if needed. For example, 45°30′15″ − 12°45′50″ requires borrowing 1 minute to turn 15 seconds into 75 seconds. After subtracting 50 seconds you have 25 seconds left, but the minutes column now has 29 minutes instead of 30, so you must borrow 1 degree to make 89 minutes. The final result is 32°44′25″.
When Surveyors and Navigators Need DMS Arithmetic
Surveyors add and subtract DMS angles constantly when reducing bearings from field notes, computing traverse angles, or applying corrections to observed directions. Navigators use DMS for celestial fixes and compass conversions. Students in geomatics, civil engineering, and geography courses also practice these calculations. Because the carry and borrow rules differ from normal base-10 arithmetic, mistakes are common — this calculator removes that friction.
Common DMS Sums and Differences
| Operation | Angle A | Angle B | Result |
|---|---|---|---|
| Add | 45°30′15″ | 12°45′50″ | 58°16′5″ |
| Subtract | 45°30′15″ | 12°45′50″ | 32°44′25″ |
| Add | 127°14′33″ | 180°0′0″ | 307°14′33″ |
| Subtract | 86°45′12″ | 90°0′0″ | −3°14′48″ |
Frequently Asked Questions
How do you add angles in degrees minutes seconds?
Add the seconds first. If the total is 60 or more, carry 1 minute and keep the remainder. Add the minutes plus any carry; if the total is 60 or more, carry 1 degree. Finally add the degrees. This calculator performs the same steps automatically and shows exactly where the carry occurred.
How do you subtract DMS when seconds are larger in the second angle?
Borrow 1 minute from the minutes column and add it as 60 seconds to the seconds column. Then subtract the seconds. If the minutes column also needs borrowing, take 1 degree and convert it to 60 minutes. The tool displays a borrow note whenever this happens.
Can minutes and seconds be negative in DMS?
In this calculator, the sign is carried by the degrees field only; minutes and seconds must be between 0 and 59. That matches the standard convention used in surveying and navigation, where an angle like −45°30′15″ means negative 45 degrees plus 30 minutes and 15 seconds in the negative direction.
Why do surveyors still use DMS instead of decimal degrees?
Historical instruments and field books use DMS, and many legal descriptions and old plats are written in DMS. Total stations can output either format, but bearings on survey drawings are traditionally labeled in degrees, minutes, and seconds because the precision is immediately visible — seconds represent roughly 30 metres at the equator, which is a meaningful field scale.
How do I convert DMS back to decimal degrees?
Divide minutes by 60, divide seconds by 3600, add both to the absolute value of the degrees, then reapply the sign from the degrees field. For example, 58°16′5″ becomes 58 + 16/60 + 5/3600 = 58.2681°. The result card on this page shows the decimal value automatically.