Layout and shop math
Degrees, minutes and seconds to decimal degrees
Decimal degrees = D + M / 60 + S / 3600; 35° 15' 30" is 35.2583°.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Decimal degrees
°
DD = D + M / 60 + S / 3600 = 35 + 15 / 60 + 30 / 3600 = 35.2583°
Degrees, minutes, seconds
D = 35; (35.258333 - 35) x 60 = 15.5 min, so M = 15; (15.5 - 15) x 60 = 30.00 sec
Radians
rad
rad = DD x pi / 180 = 35.258333 x 3.1416 / 180 = 0.615374 rad
Source: Standard angle units (1° = 60 min = 3,600 sec; pi rad = 180°)
Decimal degrees show 4 places, seconds 2 places, radians 6 places. The math uses pi to full precision; the formula line shows it as 3.1416.
How it works
DD = D + M / 60 + S / 3600; rad = DD x pi / 180
| Symbol | Meaning | Unit |
|---|---|---|
| DD | Decimal degrees | ° |
| D | Whole degrees | ° |
| M | Minutes, 1/60 of a degree | ' |
| S | Seconds, 1/3600 of a degree | " |
| rad | Radians; pi rad = 180° | rad |
Going back to D-M-S: D is the whole part of DD. Multiply what is left by 60; the whole part is M. Multiply the new remainder by 60 for S.
Worked example: D-M-S to decimal
A drawing calls 35° 15' 30".
- 15 / 60 = 0.25; 30 / 3600 = 0.00833.
- DD = 35 + 0.25 + 0.00833 = 35.2583°.
- rad = 35.2583 x 0.0174533 = 0.615374 rad.
Result: 35.2583° (0.615374 rad).
Worked example: decimal to D-M-S
A sine bar setup sheet gives 12.3456°.
- D = 12.
- 0.3456 x 60 = 20.736 min, so M = 20.
- 0.736 x 60 = 44.16 sec.
- rad = 12.3456 x 0.0174533 = 0.215471 rad.
Result: 12° 20' 44.16" (0.215471 rad).
Shop notes
- CNC controls take decimal degrees; drawings and older sine bar tables often use D-M-S.
- Negative angles: the sign applies to the whole angle. -12° 20' is -12.3333°, not -11.6667°.
- Taper half angles, sine bar angles and chamfers are where D-M-S shows up most on a drawing.
- One minute is 0.0167°; one second is 0.00028°. Round decimal degrees to 4 places for setups.
FAQ
What is 3° 34' 35" in decimal degrees?
3.5764°: 3 + 34 / 60 + 35 / 3600 = 3 + 0.56667 + 0.00972. That is the half angle of a 1.500 in/ft taper, rounded to the second.
How many degrees is one radian?
57.2958°: 180 / pi.
What is 0.5° in minutes?
30 min: 0.5 x 60.
Which form does a CNC control want?
Most controls take decimal degrees, such as 12.3456°. Convert D-M-S from the drawing first: 12° 20' 44.16" is 12.3456°.
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