Degree conversion formulae

Most GPS receivers can display the different degree formats (default format is usually D m). Hence they can be used to convert easily from one format to the other two.

InformationSmall letters (d, m, s) mean decimal numbers (e.g. 58.65375, 39.225, 13.5).Capital letters (D, M) mean integers (part of number in front of decimal point) (e.g. 58, 39).The TRUNC (truncate) function converts a decimal number to an integer by keeping the part of the number in front of the decimal point and discarding the rest(e.g. TRUNC(58.65375) = 58, TRUNC(−58.65375) = −58).Note: When converting a negative coordinate (e.g. southern or western positions) to another degree format (d, D m or D M s), let all starting values (degrees, minutes, seconds) be negative.

From decimal-degrees (d) to Degrees decimal-minutes (D m)D = TRUNC(d)m = (d − D) × 60Example: d = 58.65375°D = TRUNC(58.65375°) = 58°m = (58.65375 − 58) × 60′ = 0.65375 × 60′ = 39 . 225′ So 58.65375° corresponds to 58° 39.225′.

From decimal-degrees (d) to Degrees Minutes decimal-seconds (D M s)D = TRUNC(d)M = TRUNC((d − D) × 60)s = (d − D − M/60) × 3600 = (d − D) × 3600 − M × 60Example: d = 58.65375°D = TRUNC(58.65375°) = 58°M = TRUNC((58.65375 − 58) × 60′) = TRUNC(39.225′) = 39′s = (58.65375 − 58 − 39/60) × 3600″ = 0.0375 × 3600″ = 13 . 5″ s = (58.65375 − 58) × 3600″ − 39 × 60″ = 2353.5″ − 2340″ = 13 . 5″ So 58.65375° corresponds to 58° 39′ 13.5″.

From Degrees decimal-minutes (D m) to decimal-degrees (d)d = D + m/60Example: D = 58°, m = 39.225′d = 58° + 39.225°/60 = 58° + 0.65375° = 58 . 65375° So 58° 39.225′ corresponds to 58.65375°.

From Degrees decimal-minutes (D m) to Degrees Minutes decimal-seconds (D M s)D = D

M = TRUNC(m)s = (m − M) × 60Example: D = 58°, m = 39.225′D = 58°M = TRUNC(39.225′) = 39′s = (39.225 − 39) × 60″ = 0.225 × 60″ = 13 . 5″ So 58° 39.225′ corresponds to 58° 39′ 13.5″.

From Degrees Minutes decimal-seconds (D M s) to decimal-degrees (d)d = D + M/60 + s/3600Example: D = 58°, M = 39′, s = 13.5″d = 58° + 39°/60 + 13.5°/3600 = 58° + 0.65° + 0.00375° = 58 . 65375° So 58° 39′ 13.5″ corresponds to 58.65375°.

From Degrees Minutes decimal-seconds (D M s) to Degrees decimal-minutes (D m)D = Dm = M + s/60Example: D = 58°, M = 39′, s = 13.5″D = 58°m = 39′ + 13.5′/60 = 39′ + 0.225′ = 39 . 225′ So 58° 39′ 13.5″ corresponds to 58° 39.225′.

From an NMEA 0183 sentence (Dm,H) to Degrees decimal-minutes Hemisphere (D m H)NMEA = National Marine Electronics Association (in the US).Geographical coordinates in NMEA 0183 sentences look like this: …,Dm,H,…where H = hemisphere (N = north, S = south, E = east, W = west).Hence the above means: D° m′ H.D = TRUNC(Dm/100)m = Dm − 100 × DH = HExample: Part of an NMEA 0183 sentence containing a position: …,5839.225,N,00910.660,E,…D = TRUNC(5839.225/100) = 58°m = 5839.225 − 100 × 58 = 5839.225 − 5800 = 39 . 225′ H = N

    

D = TRUNC(00910.660/100) = 009°m = 00910.660 − 100 × 009 = 910.660 − 900 = 10 . 660′ H = E

So the position 5839.225,N,00910.660,E in an NMEA 0183 sentence equals the position 58° 39.225′ N  009° 10.660′ E.

From Degrees decimal-minutes Hemisphere (D m H) to NMEA 0183 (Dm,H) (see above too)Dm,H = 100 × D + m, HExample: D = 58°, m = 39.225′, H = NDm,H = 100 × 58 + 39.225, N = 5800 + 39.225, N = 5839 . 225,N So 58° 39.225′ N equals 5839.225,N in an NMEA 0183 sentence.

Position exampleThe coordinate formats  5839.225,N,00910.660,E  58.65375° N  009.17767° E  58° 39.225′ N  009° 10.660′ E

 58° 39′ 13.5″ N  009° 10′ 39.6″ E  all represent the same position.

More decimal-degree   (circle = 360 degrees)   conversion formulae

From decimal-degrees (d) to radians (r) (circle = 2π radians)r = d × π/180Example: d = 58.65375°r = 58.65375 radians × π/180 = 1 . 023701 radians

From decimal-degrees (d) to gons or grad(e)s (g) (circle = 400 gons or grad(e)s)g = d × 10/9Example: d = 58.65375°g = 58.65375 gons × 10/9 = 65 . 17083 gons or grad(e)s

From decimal-degrees (d) to mils (ml) (circle = 6400 mils)ml = d × 160/9Example: d = 58.65375°ml = 58.65375 mils × 160/9 = 1042 . 733 mils

From radians (r) (circle = 2π radians) to decimal-degrees (d)d = r × 180/πExample: r = 1.023701 radiansd = 1.023701° × 180/π = 58 . 65375°

From gons or grad(e)s (g) (circle = 400 gons or grad(e)s) to decimal-degrees (d)d = g × 0.9Example: g = 65.17083 gons or grad(e)sd = 65.17083° × 0.9 = 58 . 65375°

From mils (ml) (circle = 6400 mils) to decimal-degrees (d)d = ml × 9/160Example: ml = 1042.733 milsd = 1042.733° × 9/160 = 58 . 65375°