🌍 DMS ↔ Decimal Degrees Converter 2026
Convert Degrees°Minutes′Seconds″ ↔ Decimal Degrees instantly — for GPS coordinates, Google Maps, latitude/longitude, astronomy, surveying & navigation. Includes structured DMS input, batch converter, precision table, Haversine formula & WGS 84 explained.
🔄 DMS ↔ Decimal Degrees Converter
Enter degrees, minutes, and seconds in separate fields — or paste a full DMS string.
⚡ Quick examples — click to load:
Enter a decimal degree value. Use negative for South/West.
⚡ Quick examples — click to load:
One coordinate per line. Mix DMS and decimal — the converter auto-detects each format.
📖 How to Use This DMS Converter
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1DMS → Decimal: Enter Degrees, Minutes, Seconds
Use separate number fields for each component, or paste a full DMS string in any common format (40° 26′ 46″, "40 26 46 N", "40d26m46s"). Click a quick example to load sample data instantly.
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2Decimal → DMS: Enter a Decimal Number
Switch to the "Decimal → DMS" tab. Enter any decimal degree value — positive (N/E) or negative (S/W). Select Latitude or Longitude for correct direction labelling. The result shows DMS format and multiple output styles.
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3Batch Convert Multiple Coordinates
Use the Batch tab to paste multiple coordinates (mix of DMS and decimal) — one per line. Get all results at once, then copy to clipboard for use in spreadsheets, GIS software, or databases.
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4Read the Result Panel
The blue result box shows the converted value in multiple formats: decimal degrees (6 places), DMS with symbols, DMS with spaces only, radians, and distance-per-arcsecond at the equator.
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5Copy & Use in Google Maps, APIs, or GIS
Click "Copy" to copy the result. For Google Maps: paste as "lat, lon" in decimal format. For GeoJSON: use decimal degrees. For traditional maps: paste DMS format.
📐 DMS ↔ Decimal Degrees — Complete Mathematical Formulas
\( \theta_{\text{dec}} = D + \frac{M}{60} + \frac{S}{3600} \qquad \text{(positive/North/East)} \)
\( \theta_{\text{dec}} = -\!\left(D + \frac{M}{60} + \frac{S}{3600}\right) \qquad \text{(South/West, always negate after summing)} \)
\( \text{Example: } 40°\,26'\,46'' = 40 + \frac{26}{60} + \frac{46}{3600} = 40 + 0.4\overline{3} + 0.012\overline{7} = \mathbf{40.44611\overline{1}°} \)
\( D = \lfloor |\theta_{\text{dec}}| \rfloor \qquad \text{(integer part)} \)
\( M_{\text{float}} = (|\theta_{\text{dec}}| - D) \times 60 \qquad M = \lfloor M_{\text{float}} \rfloor \)
\( S = (M_{\text{float}} - M) \times 60 \)
\( \text{Sign: if } \theta_{\text{dec}} < 0 \text{, then South (latitude) or West (longitude)} \)
\( \text{Example: } 40.446111° \to D=40,\; (0.446111)\times60=26.7667,\; M=26,\; (0.7667)\times60=46.00''\)
DMS → Decimal (London): 51°30′26.3″ N → \(51 + 30/60 + 26.3/3600 = 51 + 0.5 + 0.007305... = 51.507305°\,\text{N}\)
DMS → Decimal (Sydney): 33°51′54″ S → \(-(33 + 51/60 + 54/3600) = -(33 + 0.85 + 0.015) = -33.865°\)
Decimal → DMS (San Francisco): −122.4194° → D=122, (0.4194×60)=25.164, M=25, (0.164×60)=9.84″ → 122°25′9.84″ W
Decimal → DMS (GPS minute): 0.016667° (1 arcminute) → D=0, (0.016667×60)=1.0, M=1, S=0 → 0°1′0″
📏 Coordinate Precision — How Many Decimal Places Do You Need?
\( 1° \approx 111{,}320\,\text{m} \approx 111.32\,\text{km} \qquad \text{(at equator, WGS 84)} \)
\( 1' = 1°/60 \approx 1{,}855.3\,\text{m} \approx 1.855\,\text{km} = 1\,\text{nautical mile} \)
\( 1'' = 1°/3600 \approx 30.92\,\text{m} \approx 31\,\text{m} \)
\( 0.1° \approx 11{,}132\,\text{m} \quad 0.01° \approx 1{,}113\,\text{m} \quad 0.001° \approx 111\,\text{m} \)
\( 0.0001° \approx 11.1\,\text{m} \quad 0.00001° \approx 1.11\,\text{m} \quad 0.000001° \approx 11\,\text{cm} \)
\( \text{At latitude } \varphi:\; d_{\text{lon}} = 111{,}320 \times \cos(\varphi)\,\text{m/degree} \)
| Format | Precision | Ground Distance (equator) | Use Case |
|---|---|---|---|
| 1° (nearest degree) | 1° | ≈ 111 km | Country/region level |
| 1° 1′ (DMS min) | 1′ | ≈ 1.855 km | City district level |
| 1° 1′ 1″ (DMS sec) | 1″ | ≈ 31 m | Street / building level |
| 0.1° (1 decimal) | ±0.05° | ≈ 5.6 km | Town level |
| 0.01° (2 decimals) | ±0.005° | ≈ 556 m | Neighbourhood |
| 0.001° (3 decimals) | ±0.0005° | ≈ 55.6 m | Street |
| 0.0001° (4 decimals) | ±0.00005° | ≈ 5.56 m | Individual building |
| 0.00001° (5 decimals) | ±0.000005° | ≈ 1.11 m | Consumer GPS typical |
| 0.000001° (6 decimals) | ±0.0000005° | ≈ 11 cm | Survey/RTK GPS |
| 0.0000001° (7 decimals) | — | ≈ 1.1 cm | Geodetic benchmarks |
| DMS decimal: 1° 1′ 1.0″ | 0.1″ | ≈ 3.09 m | High-precision survey |
| DMS decimal: 1° 1′ 1.00″ | 0.01″ | ≈ 30.9 cm | Millimetre-level work |
📡 Haversine Formula — Distance Between Two GPS Points
\( a = \sin^2\!\left(\frac{\Delta\varphi}{2}\right) + \cos\varphi_1 \cdot \cos\varphi_2 \cdot \sin^2\!\left(\frac{\Delta\lambda}{2}\right) \)
\( c = 2\,\operatorname{atan2}\!\left(\sqrt{a},\,\sqrt{1-a}\right) \qquad d = R \cdot c \)
\( R = 6{,}371{,}009\,\text{m} \text{ (mean Earth radius, WGS 84)} \)
\( \Delta\varphi = \varphi_2 - \varphi_1 \quad \Delta\lambda = \lambda_2 - \lambda_1 \quad \text{(all in radians)} \)
🏛️ History — Why Do We Use Degrees, Minutes & Seconds?
The degrees-minutes-seconds system is one of humanity's oldest surviving measurement conventions. It originates with the Babylonians, who used a base-60 (sexagesimal) numerical system around 2000 BCE. Unlike our base-10 system, base-60 has an appeal that base-10 lacks: the number 60 divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 — making fractions enormously more convenient without the need for decimal notation.
The astronomer Hipparchus (c. 190–120 BCE) is credited with dividing the circle into 360 degrees, each degree into 60 arcminutes, and each arcminute into 60 arcseconds. Ptolemy (c. 100–170 CE) standardised this system in the Almagest and Geographia, creating the first world maps using latitude and longitude in DMS notation. No subsequent civilisation felt the need to change this system, and so it survived the fall of Rome, the Islamic Golden Age, the Age of Exploration, and into the satellite era.
Why didn't the metric system replace DMS? The French Revolution did attempt this: French scientists proposed a "decimal degree" grade system (100 grades per right angle, 400 per circle — the modern gradian). It was briefly used in France (c. 1794–1800) then abandoned because the existing astronomical tables, navigational charts, and surveying instruments all used degrees. Inertia won. Today, decimal degrees are the numerical format standard in computing and GPS — but they are still degrees, not a truly metric angle unit.
🛰️ GPS Coordinates, WGS 84 & Decimal Degrees in Modern Systems
WGS 84 (World Geodetic System 1984) is the reference coordinate system used by GPS and Google Maps. It models the Earth as an oblate spheroid (ellipsoid) with equatorial radius \(a = 6{,}378{,}137.0\,\text{m}\) and polar radius \(b = 6{,}356{,}752.314\,\text{m}\), giving a flattening \(f = (a-b)/a \approx 1/298.257\). All GPS satellite coordinates are broadcast in WGS 84, and your phone converts these to latitude/longitude in decimal degrees.
Consumer GPS (Smartphone)
Accuracy: ±3–5 m (open sky), ±10–20 m (urban canyon). Equivalent precision: 5 decimal places (±1.11 m). Most mapping apps store 6 decimal places. The iPhone 15/Galaxy S24 combine GPS with Wi-Fi positioning (WPS) to achieve ±1–3 m in cities.
DGPS — Differential GPS
Accuracy: ±0.1–1 m. Uses fixed reference stations to broadcast real-time corrections. Used in surveying, marine navigation (IALA Differential GPS network), and precision farming. Requires a DGPS-capable receiver and proximity to a correction station.
RTK GPS (Real-Time Kinematic)
Accuracy: ±1–2 cm. Resolves carrier phase ambiguities using a base station + rover pair. Used in precision agriculture, construction staking, autonomous vehicles, and geodetic control surveys. Requires 6+ decimal places (7–8 for cm-level).
Google Maps / APIs
Google Maps uses decimal degrees internally (WGS 84). The URI format: https://www.google.com/maps?q=51.5072,-0.1276. Google Maps JavaScript API accepts both DD and DMS via LatLng objects. OpenStreetMap, Mapbox, Leaflet.js, and ArcGIS Online all use decimal degrees as primary format.
🔭 Astronomy — Right Ascension, Declination & Hour Angles
In astronomy, the sky is mapped using a coordinate system analogous to latitude/longitude. Declination (Dec) is the sky equivalent of latitude, expressed in degrees, arcminutes, arcseconds — identical in format to geographic DMS. Polaris (North Star) lies at Dec = +89°15′50.8″. The celestial equator is Dec = 0°.
Right Ascension (RA) is the sky equivalent of longitude, but measured in hours, minutes, seconds of time (not angle). The full circle = 24 hours = 360°, so 1 hour RA = 15°, 1 minute RA = 0.25° = 15 arcminutes, 1 second RA = 15 arcseconds. The Sun moves approximately 1° per day along the ecliptic = 4 minutes of RA per day.
\( \alpha_{\text{deg}} = H \times 15° + M_{\text{time}} \times 0.25° + S_{\text{time}} \times (0.25°/60) \qquad \text{(RA hours→degrees)} \)
\( 1\,\text{hour RA} = 15° \qquad 1\,\text{min RA} = 15' \qquad 1\,\text{sec RA} = 15'' \)
\( \text{Dec (DMS)} = \pm D° \, M' \, S'' \qquad \text{(same formula as geographic DMS→decimal)} \)
\( \text{Example: Sirius RA} = 6^h 45^m 8.9^s = 6\times15+45\times0.25+8.9/240 = 101.29°\)
❓ Frequently Asked Questions — DMS & Decimal Degrees
How do I convert degrees minutes seconds to decimal degrees?
How do I convert decimal degrees to DMS?
What does 40° 26' 46" mean in decimal degrees?
How do I enter coordinates in Google Maps?
51.5072, -0.1276 (no quotes) — this opens London, UK. Use a comma between latitude and longitude. Negative latitude = South; negative longitude = West. Google Maps also accepts DMS format: 51°30'26.0"N 0°7'39.4"W but decimal is more reliable and easier to paste.