Convert one angle into degrees, radians, gradians, turns, arcminutes, and arcseconds all at once, with a DMS check to catch unit mix-ups.
Table of contents
How to use our Angle Conversion Calculator
- Enter the number in Angle value.
- Choose the matching unit in Convert from, such as degrees, radians, or arcseconds.
- Pick Decimal places to control how rounded the displayed results look.
- Click Calculate to see Converted value, all common angle units, and the Degree-minute-second form.
- Sanity-check the answer using Quick meaning and the unit list: for example, 90 degrees should match 0.25 turns and about 1.570796 radians.
Definitions
Angle value: The starting amount you want to convert.
Convert from: The unit attached to your starting number.
Decimal places: How many digits appear after the decimal in displayed results.
Degrees (deg): A full circle is 360 degrees.
Radians (rad): A standard math and science angle unit; a full circle is 2 pi radians.
Gradians (grad): An angle unit where a right angle is 100 grad and a full circle is 400 grad.
Turns (turn): Part of one full rotation; 1 turn means one complete circle.
Arcminutes (arcmin): Small angle units equal to 1/60 of a degree.
Arcseconds (arcsec): Smaller angle units equal to 1/3600 of a degree.
Degree-minute-second form: A way to write decimal degrees as whole degrees, whole arcminutes, and remaining arcseconds.
Common mistakes and quick fixes
Mistake: Typing 30 in Angle value but leaving Convert from on radians when you really mean 30 degrees.
Fix: Make sure Convert from matches the unit you started with before you calculate.
Mistake: Reading Degree-minute-second form as if decimal degrees worked like normal decimals, such as thinking 12.30 degrees means 12 degrees 30 minutes.
Fix: Use Degree-minute-second form to see the true breakdown; 12.30 degrees is 12 degrees, 18 minutes, 0 seconds.
Mistake: Confusing Arcminutes with minutes of time, or Arcseconds with seconds of time.
Fix: Treat Arcminutes and Arcseconds as angle units only: 1 degree = 60 arcminutes and 3600 arcseconds.
Mistake: Choosing too few Decimal places and thinking a tiny result is exactly zero.
Fix: Increase Decimal places if the converted value is very small, especially in Turns or Radians .
Mistake: Expecting negative input in Angle value to become positive in the outputs.
Fix: Negative angles stay negative in Degrees , Radians , Turns , and the other unit results.
Mistake: Entering a blank or non-number in Angle value and expecting a conversion.
Fix: Enter a real number in Angle value , such as 45, 1.5708, or -30, then calculate again.
Limitations & Key Assumptions / Boundary Conditions
- Results are mathematical unit conversions only. They do not choose a direction convention for your class, graph, or physics problem beyond keeping the sign you entered.
- Displayed values are rounded to the selected Decimal places, so the screen may hide tiny nonzero differences even though the internal conversion uses higher precision.
- Degree-minute-second form is a supporting interpretation based on decimal degrees. It is helpful for checking understanding, but it is not the main input format here.
- Very large inputs can lose some precision because browser math uses finite floating-point numbers.
- The calculator converts only these common units: degrees, radians, gradians, turns, arcminutes, and arcseconds. It does not include less common angle units.
Methodology
How the calculator converts angles
The calculator first changes your starting value into turns, which means full rotations. Then it converts that turn value into every other common angle unit. Using turns as the base keeps the relationships simple and exact for the included units.
turn = value / unit_per_turn
deg = turn * 360
rad = turn * 2*pi
grad = turn * 400
arcmin = deg * 60
arcsec = deg * 3600
Degree-minute-second breakdown
After finding decimal degrees, the calculator builds a degree-minute-second result so you can compare decimal degrees with DMS style.
D = trunc(abs(deg))
M_full = (abs(deg) - D) * 60
M = trunc(M_full)
S = (M_full - M) * 60
If rounding makes seconds reach 60, the calculator carries 1 minute forward. If minutes then reach 60, it carries 1 degree forward too. The original negative sign is kept on the full DMS value.
Mini example
Suppose Angle value is 45 and Convert from is degrees. First, turns = 45 / 360 = 0.125. Then radians = 0.125 * 2*pi = about 0.785398, gradians = 0.125 * 400 = 50, arcminutes = 45 * 60 = 2700, and arcseconds = 45 * 3600 = 162000. The Quick meaning for 45 degrees is one eighth of a full turn.
Assumptions used
The calculator assumes standard angle relationships:
1 turn = 360 degrees, 1 turn = 2 pi radians, 1 turn = 400 gradians, 1 degree = 60 arcminutes, and 1 degree = 3600 arcseconds. Results may look slightly different from hand work if you round at different steps.