acalculator

Degree calculator: DMS and decimal

Type an angle in degrees, minutes and seconds, or in decimal degrees. The degree calculator converts it both ways, gives radians, and adds or subtracts two angles.

Your numbers

Do
Decimal degrees
12.5125

The angle is 12.5125°, or 12° 30′ 45″.

Degrees, minutes, seconds
12° 30′ 45″
Radians
0.218384322656
Between 0° and 360°
12.5125
Total arcseconds
45,045

Decimal degrees: 12.5125. The angle is 12.5125°, or 12° 30′ 45″.

How to calculate

Converts an angle between degrees, minutes and seconds (DMS) and decimal degrees, with radians, and adds or subtracts two angles in DMS.

Example with the default inputs (Do Convert, Degrees 12, Minutes (′) 30, Seconds (″) 45): The angle is 12.5125°, or 12° 30′ 45″.

Method: Decimal degrees = d + m ÷ 60 + s ÷ 3,600 (minutes and seconds take the sign of d); radians = degrees × π ÷ 180; sums and differences are taken in arcseconds.

  • Minutes and seconds are from 0 to less than 60 and take the sign of the degrees: −12° 30′ is −12.5°. For an angle between −1° and 0°, type negative decimal degrees.
  • Seconds are shown to at most 4 decimals, rounded half up, with a rounded 60″ carried into the minutes.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Do Convert, Degrees 12, Minutes (′) 30, Seconds (″) 45 gives Decimal degrees 12.5125, Degrees, minutes, seconds 12° 30′ 45″, Total arcseconds 45,045.Source: NIST SP 330 (2019), §4 Non-SI units accepted for use with the SI (1° = π/180 rad; 1′ = 1/60°; 1″ = 1/60′). https://www.nist.gov/pml/special-publication-330/sp-330-section-4
  2. Do Convert, Degrees 40.446195 gives Degrees, minutes, seconds 40° 26′ 46.302″, Decimal degrees 40.446195.Source: NIST SP 330 (2019), §4 Non-SI units accepted for use with the SI (1° = π/180 rad; 1′ = 1/60°; 1″ = 1/60′). https://www.nist.gov/pml/special-publication-330/sp-330-section-4
  3. Do Add, Degrees 12, Minutes (′) 30, Seconds (″) 45, Second angle: degrees 5, Second angle: minutes (′) 45, Second angle: seconds (″) 30 gives Degrees, minutes, seconds 18° 16′ 15″, Decimal degrees 18.270833.Source: NIST SP 330 (2019), §4 Non-SI units accepted for use with the SI (1° = π/180 rad; 1′ = 1/60°; 1″ = 1/60′). https://www.nist.gov/pml/special-publication-330/sp-330-section-4
  4. Do Subtract, Degrees 10, Minutes (′) 0, Seconds (″) 0, Second angle: degrees 25, Second angle: minutes (′) 30, Second angle: seconds (″) 0 gives Degrees, minutes, seconds −15° 30′ 0″, Decimal degrees -15.5, Between 0° and 360° 344.5.Source: OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (radians = degrees × π ÷ 180; coterminal angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (coterminal angles: add 360°)
  5. Do Convert, Degrees 800 gives Between 0° and 360° 80, Radians 13.962634.Source: OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (radians = degrees × π ÷ 180; coterminal angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (800° is coterminal with 80°)
  6. Do Convert, Degrees 15 gives Radians 0.261799.Source: OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (radians = degrees × π ÷ 180; coterminal angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (15° = π/12)

How it works

  • An angle in arcseconds: d° m′ s″ = 3,600 d + sign(d) × (60 m + s), where sign(d) is −1 when the degrees are below 0 and +1 otherwise (0 and −0 count as +1). Empty minutes or seconds count as 0.
  • Add or subtract: first angle ± second angle, in arcseconds.
  • Decimal degrees = arcseconds ÷ 3,600.
  • DMS: the size of the arcseconds is rounded to 4 decimals (halves up), then split into whole degrees (÷ 3,600), whole minutes (÷ 60 of what is left) and seconds; so 59.99995″ carries to the next minute. A minus sign goes in front when the angle is negative and does not round to 0.
  • Radians = decimal degrees × π ÷ 180 (double precision).
  • Between 0° and 360° = degrees − 360 × ⌊degrees ÷ 360⌋.

Every step except radians is an exact fraction of the typed decimals. Decimal degrees, radians and arcseconds show to 12 significant figures, rounded half up.

Assumptions

  • Degrees from −1,000,000,000 to 1,000,000,000; minutes and seconds from 0 to less than 60.
  • One degree is 60 minutes of arc and one minute is 60 seconds of arc (NIST SP 330).

Worked examples by hand

12° 30′ 45″. 12 + 30 ÷ 60 + 45 ÷ 3,600 = 12.5125° = 45,045″.

40.446195°. 0.446195 × 60 = 26.7717′; 0.7717 × 60 = 46.302″: 40° 26′ 46.302″.

12° 30′ 45″ + 5° 45′ 30″. 45″ + 30″ = 75″ = 1′ 15″; 30′ + 45′ + 1′ = 76′ = 1° 16′; 12° + 5° + 1° = 18° 16′ 15″ = 18.2708333°.

10° − 25° 30′. 10 − 25.5 = −15.5° = −15° 30′ 0″; between 0° and 360°: −15.5 + 360 = 344.5°.

800° (OpenStax). 800 − 2 × 360 = 80°; 800 × π ÷ 180 = 13.9626 rad.

15° (OpenStax). 15 × π ÷ 180 = π/12 = 0.261799 rad.

Other questions people ask

How do I convert DMS to decimal degrees?

Add the degrees, the minutes ÷ 60 and the seconds ÷ 3,600. 12° 30′ 45″ = 12 + 0.5 + 0.0125 = 12.5125°.

How do I convert decimal degrees to DMS?

The whole number is the degrees. Multiply the decimal part by 60: the whole number is the minutes. Multiply what is left by 60 for the seconds. 40.446195° = 40° and 0.446195 × 60 = 26.7717′, then 0.7717 × 60 = 46.302″.

How do I add angles in degrees, minutes and seconds?

Add the seconds, the minutes and the degrees separately, then carry: every 60″ is 1′ and every 60′ is 1°. 12° 30′ 45″ + 5° 45′ 30″ = 17° 75′ 75″ = 18° 16′ 15″.

How do I convert degrees to radians?

Multiply by π ÷ 180. 15° = 15π ÷ 180 = π/12 ≈ 0.2618 rad, and 180° = π rad.

What does "between 0° and 360°" mean?

Angles that differ by whole turns of 360° point the same way (coterminal angles). 800° − 2 × 360° = 80°, and −15.5° + 360° = 344.5°.

How do I type a negative angle?

Make the degrees negative; the minutes and seconds count the same way, so −12° 30′ is −12.5°. For an angle between −1° and 0°, type it as negative decimal degrees, such as −0.5.