Degrees, minutes, and seconds are a way to measure angles with more precision than whole degrees alone. This system is often called DMS notation, and it splits one degree into 60 minutes and one minute into 60 seconds. It is useful when a small change in angle matters, such as in maps, navigation, surveying, astronomy, and geometry.
Learning DMS helps connect circle geometry with real-world coordinate and direction measurements.
DMS works like a base-60 place-value system for angles. To convert DMS to decimal degrees, minutes are divided by 60 and seconds are divided by 3600, then added to the degrees. To convert decimal degrees to DMS, keep the whole degrees, multiply the decimal part by 60 to get minutes, then multiply the remaining decimal part by 60 to get seconds.
For example, 42° 18′ 36″ = 42 + 18/60 + 36/3600 = 42.31°.
Key Facts
- 1 full circle = 360°
- 1° = 60′
- 1′ = 60″
- 1° = 3600″
- DMS to decimal degrees: decimal degrees = degrees + minutes/60 + seconds/3600
- Decimal degrees to DMS: degrees are the whole number, minutes = decimal part × 60, seconds = remaining decimal part × 60
Vocabulary
- Degree
- A degree is a unit of angle measure equal to 1/360 of a full rotation.
- Minute
- A minute of angle is 1/60 of a degree and is written with the symbol ′.
- Second
- A second of angle is 1/60 of a minute and is written with the symbol ″.
- DMS notation
- DMS notation writes an angle using degrees, minutes, and seconds, such as 35° 12′ 20″.
- Decimal degree
- A decimal degree writes an angle as a single decimal number of degrees, such as 35.2056°.
Common Mistakes to Avoid
- Treating minutes and seconds like base-10 decimals is wrong because DMS uses base 60, so 30′ means half a degree, not 0.30°.
- Forgetting to divide seconds by 3600 is wrong because seconds are 1/3600 of a degree, not 1/60 of a degree.
- Writing 75° 80′ 10″ as a final DMS angle is wrong because minutes and seconds should usually be regrouped so each is less than 60.
- Rounding too early during conversion is wrong because it can change the final angle, so keep extra digits until the last step.
Practice Questions
- 1 Convert 28° 45′ 30″ to decimal degrees.
- 2 Convert 63.275° to degrees, minutes, and seconds.
- 3 A survey map gives a direction in DMS instead of decimal degrees. Explain why DMS notation can be useful when measuring or recording very precise angles.