Degrees to Minutes, Seconds Converter
Convert between decimal degrees and degrees, minutes, seconds (DMS) format or Degree Converter.
Decimal Degrees → DMS
DMS → Decimal Degrees
How Degrees, Minutes, Seconds (DMS) Work
Degrees, minutes, and seconds (DMS) is a system for expressing angles and geographic coordinates. One full circle is 360 degrees. Each degree is divided into 60 minutes (denoted by ′), and each minute is divided into 60 seconds (denoted by ″). This gives 3,600 seconds per degree. DMS notation is widely used in navigation, surveying, cartography, and astronomy. For example, the coordinates of the Statue of Liberty are approximately 40° 41′ 21″ N, 74° 2′ 40″ W.
DMS Conversion Formulas
Decimal Degrees to DMS: Degrees = integer part of the decimal. Minutes = integer part of (fractional part × 60). Seconds = remainder × 60. In formula form: d = floor(DD), m = floor((DD − d) × 60), s = ((DD − d) × 60 − m) × 60. DMS to Decimal Degrees: DD = d + m/60 + s/3600. For negative angles (south latitude or west longitude), apply the sign to the entire result.
Worked Example: Convert 45.5125° to DMS
Step 1: The whole number part is the degrees: 45°. Step 2: Take the decimal part: 0.5125. Multiply by 60: 0.5125 × 60 = 30.75. The whole number part is the minutes: 30′. Step 3: Take the remaining decimal: 0.75. Multiply by 60: 0.75 × 60 = 45. The seconds: 45″. Result: 45.5125° = 45° 30′ 45″. Reverse check: 45 + 30/60 + 45/3600 = 45 + 0.5 + 0.0125 = 45.5125° ✓.
Common Angle Conversions
0° = 0° 0′ 0″ · 30° = 30° 0′ 0″ · 45° = 45° 0′ 0″ · 45.5° = 45° 30′ 0″ · 60° = 60° 0′ 0″ · 90° = 90° 0′ 0″ · 120° = 120° 0′ 0″ · 180° = 180° 0′ 0″ · 270° = 270° 0′ 0″ · 360° = 360° 0′ 0″. GPS examples: 40.7128° N = 40° 42′ 46.08″ N (New York City) · 51.5074° N = 51° 30′ 26.64″ N (London) · 35.6762° N = 35° 40′ 34.32″ N (Tokyo).
Technical Details
DMS is a sexagesimal (base-60) subdivision system inherited from Babylonian mathematics. Minutes and seconds in angle measurement are distinct from minutes and seconds of time, though they share the same base-60 structure. Precision: 1 second of latitude equals approximately 30.87 meters on Earth's surface. 1 second of longitude varies with latitude — at the equator it is about 30.87 meters, decreasing to 0 at the poles. This converter handles negative angles for south latitudes and west longitudes, and displays seconds to 4 decimal places for sub-meter GPS accuracy.
Frequently Asked Questions
What is 90.5 degrees in DMS? 90° 30′ 0″. The 0.5 degree equals 30 minutes (0.5 × 60 = 30). How accurate is DMS notation? With seconds expressed to two decimal places, DMS can represent positions to within about 0.3 meters — sufficient for most surveying and navigation. Why are there 60 minutes in a degree? This convention comes from the ancient Babylonian base-60 (sexagesimal) number system, which was adopted by Greek astronomers and has been used ever since. What is the difference between decimal degrees and DMS? They represent the same angle in different formats. Decimal degrees (45.5125°) are easier for calculations, while DMS (45° 30′ 45″) is traditional in navigation and map reading. Can DMS values be negative? Yes. Negative values indicate south latitude or west longitude in geographic coordinates, or clockwise rotation in some engineering contexts.
Solved Examples
Example 1: A GPS reading shows latitude 33.8688° S. Convert to DMS. Degrees: 33°. Minutes: 0.8688 × 60 = 52.128, so 52′. Seconds: 0.128 × 60 = 7.68″. Answer: 33° 52′ 7.68″ S (this is Sydney, Australia). Example 2: Convert 72° 15′ 30″ to decimal degrees. DD = 72 + 15/60 + 30/3600 = 72 + 0.25 + 0.00833 = 72.2583°. Answer: 72.2583°. Example 3: A surveyor measures an angle of 128.7542°. Express in DMS. Degrees: 128°. 0.7542 × 60 = 45.252 → 45′. 0.252 × 60 = 15.12″. Answer: 128° 45′ 15.12″. Example 4: An astronomy catalog lists a star at declination −23° 26′ 21″. Convert to decimal. DD = −(23 + 26/60 + 21/3600) = −(23 + 0.4333 + 0.00583) = −23.4392°. Answer: −23.4392°.
Practice Questions
1. Convert 60.505° to DMS format. (Answer: 60° 30′ 18″) 2. Express 25° 45′ 0″ in decimal degrees. (Answer: 25.75°) 3. Convert 100.125° to DMS. (Answer: 100° 7′ 30″) 4. What is 40° 26′ 46″ as decimal degrees? (Answer: 40.4461°) 5. Convert −73.9857° to DMS. (Answer: −73° 59′ 8.52″) 6. A land survey shows bearing N 45° 30′ 0″ E. What is this in decimal degrees? (Answer: 45.5°)
Common Mistakes to Avoid
The most common error is treating minutes and seconds as decimals of 100 rather than base-60 values. For example, 45.30 degrees does NOT equal 45° 30′ — it actually equals 45° 18′ (0.30 × 60 = 18). Another frequent mistake is forgetting that seconds can exceed 59.999 due to rounding — if your calculation gives 60 seconds, carry it to the next minute. Similarly, 60 minutes must be carried to the next degree. Students often confuse the conversion direction: to go FROM DMS to decimal, divide minutes by 60 and seconds by 3600 (don't multiply). When working with negative coordinates (south or west), apply the negative sign to the entire decimal degree result, not to individual components. Finally, mixing up latitude and longitude ranges causes issues — latitude goes from −90° to +90°, while longitude goes from −180° to +180°.
Key Takeaways
DMS divides each degree into 60 minutes and each minute into 60 seconds (3600 seconds per degree). Decimal to DMS: extract whole degrees, multiply remainder by 60 for minutes, multiply remainder by 60 for seconds. DMS to Decimal: degrees + minutes/60 + seconds/3600. 1 arc-second ≈ 30.87 meters on Earth's surface at the equator. DMS notation is standard for navigation, aviation charts, and topographic maps. Decimal degrees are preferred for mathematical calculations and computer applications. The system originated from Babylonian base-60 mathematics over 4,000 years ago.