# Degree calculator: DMS and decimal

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

- Page: https://www.acalculator.org/math/degree-calculator
- JSON spec: https://www.acalculator.org/math/degree-calculator.json
- Version: 28487b5e2268

## Default answer

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

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | Do | Convert one angle, or add or subtract two angles. |
| d | Degrees | The whole or decimal degrees of the angle. Negative for a negative angle. |
| m | Minutes (′) | The minutes of arc, from 0 to less than 60. Leave empty for none. |
| s | Seconds (″) | The seconds of arc, from 0 to less than 60. Leave empty for none. |
| d2 | Second angle: degrees | The degrees of the angle to add or subtract. |
| m2 | Second angle: minutes (′) | The minutes of the second angle. |
| s2 | Second angle: seconds (″) | The seconds of the second angle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| decimal | Decimal degrees | The angle in degrees: degrees + minutes ÷ 60 + seconds ÷ 3,600. |
| dms | Degrees, minutes, seconds | The angle as d° m′ s″, seconds to at most 4 decimals. |
| radians | Radians | Decimal degrees × π ÷ 180. |
| normalized | Between 0° and 360° | The coterminal angle from 0° up to 360°, in decimal degrees. |
| arcsec | Total arcseconds | The angle in seconds of arc: decimal degrees × 3,600. |

## 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.

## Assumptions

- 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.

## Worked examples

1. mode = convert, d = 12, m = 30, s = 45 gives decimal = 12.5125, dms = 12° 30′ 45″, arcsec = 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. mode = convert, d = 40.446195 gives dms = 40° 26′ 46.302″, decimal = 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. mode = add, d = 12, m = 30, s = 45, d2 = 5, m2 = 45, s2 = 30 gives dms = 18° 16′ 15″, decimal = 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. mode = subtract, d = 10, m = 0, s = 0, d2 = 25, m2 = 30, s2 = 0 gives dms = −15° 30′ 0″, decimal = -15.5, normalized = 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. mode = convert, d = 800 gives normalized = 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. mode = convert, d = 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).

## FAQ

### 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.

## Sources

- NIST SP 330 (2019), §4 Non-SI units accepted for use with the SI (degree 1° = π/180 rad; minute 1′ = 1/60°; second 1″ = 1/60′). https://www.nist.gov/pml/special-publication-330/sp-330-section-4 (retrieved 2026-10-02)
- OpenStax, Algebra and Trigonometry 2e, §7.1 Angles (degrees and radians, θ ÷ 180 = θ_R ÷ π; coterminal angles: 800° is coterminal with 80°). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles (retrieved 2026-10-02)