Angle degrees to decimal

Calculator and formula to convert angle degrees to decimal

Angle calculator

Angle format conversion

This calculator converts between decimal degrees and degrees, minutes and seconds (DMS) in both directions.

What would you like to calculate?
DMS → Decimal degrees
Decimal degrees → DMS
Angle in Degrees - Minutes - Seconds
°
Whole degrees
'
Arc minutes
''
Arc seconds
Conversion result
Decimal degrees:
Degrees:
Minutes:
Seconds:
1° = 60' = 3600'' | 1' = 60''

Angle format info

Angle format properties

Conversion: Between decimal degrees and degrees-minutes-seconds

1° = 60' 1' = 60'' 1° = 3600''

DMS format: Degrees° Minutes' Seconds''
Decimal: Degrees with fractional part

Examples
90.5° = 90° 30' 0''
45° 30' 45'' = 45.5125°
180° 0' 0'' = 180.0°

Angle format conversion: precision across representations

Converting between angle formats allows flexible representation and precise calculations:

  • Decimal degrees: Modern, computer-friendly representation
  • DMS format: Traditional degrees-minutes-seconds system
  • Precision: Up to 10 decimal places possible
  • Navigation: GPS and surveying use both formats
  • Astronomy: High-precision angle measurements
  • Surveying: Professional land surveying

Formulas for angle conversion

Degrees, minutes, seconds → Decimal
\[\text{decimal} = \text{degrees} + \frac{\text{minutes}}{60} + \frac{\text{seconds}}{3600}\]

Convert to decimal degrees

Decimal → Degrees, minutes, seconds
\[\text{degrees} = \lfloor\text{decimal}\rfloor\] \[x = (\text{decimal} - \text{degrees}) \times 60\] \[\text{minutes} = \lfloor x \rfloor\] \[\text{seconds} = (x - \text{minutes}) \times 60\]

Step-by-step decomposition into DMS format

Base units
\[1° = 60' = 3600''\] \[1' = 60''\]

Fundamental conversion factors

Floor function
\[\lfloor x \rfloor = \text{greatest integer} \leq x\]

Rounding down to the nearest integer

Examples for angle conversion

Example 1: DMS → Decimal
50° 30' 45''
\[\text{decimal} = 50 + \frac{30}{60} + \frac{45}{3600}\] \[= 50 + 0.5 + 0.0125\] \[= 50.5125°\]

Convert to decimal degrees

Example 2: Decimal → DMS
75.75°
\[\text{degrees} = \lfloor 75.75 \rfloor = 75°\] \[x = (75.75 - 75) \times 60 = 45\] \[\text{minutes} = \lfloor 45 \rfloor = 45'\] \[\text{seconds} = (45 - 45) \times 60 = 0''\]

Result: 75° 45' 0''

Practical applications
GPS: 52.5200° N
52° 31' 12'' N
Navigation: 13.4050° E
13° 24' 18'' E
Astronomy: 90° 0' 0''
90.0000°

Real coordinates and angle measurements

Applications of angle conversion

Angle format conversion is essential in many technical and scientific fields:

Navigation & GPS
  • GPS coordinates in different formats
  • Nautical and aviation navigation
  • Cartography and surveying
  • Surveying engineering
Astronomy & Space
  • Celestial coordinates
  • Telescope control
  • Satellite orbit calculations
  • Planetary missions
Engineering
  • CAD software and technical drawings
  • Robotics and automation
  • Machine control
  • Optical instruments
Education & Research
  • Mathematics and physics teaching
  • Scientific calculations
  • Data analysis and statistics
  • Laboratory instruments

Angle formats: precision and versatility in measurement

Converting between angle formats is a fundamental tool in metrology, navigation and scientific computation. Two main systems - decimal degrees and the traditional degrees-minutes-seconds format - offer different advantages for various applications, from GPS navigation to high-precision astronomy.

Summary

Angle format conversion combines practical necessity with mathematical elegance. Simple formulas - decimal = degrees + min/60 + sec/3600 - enable precise transformations between both systems. From modern GPS navigation to traditional maritime navigation and high-precision astronomy, this conversion remains an indispensable tool. It links the computer-friendly decimal representation with the intuitive DMS notation and shows how different mathematical representations describe the same physical angle precisely.

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