Angle Converter: Degrees to Radians, Gradians, Turns and Degrees Minutes Seconds, with Arc Length

Convert an angle between every unit, and get its trig values and arc length.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the unit your angle is in and enter it. For degrees-minutes-seconds fill in the three boxes.

Enter a radius if you want the arc length and sector area.

The calculator gives the angle in all the units, as a fraction of pi, and its sine, cosine and tangent.

Formulas used

  • \theta_{rad} = \theta^\circ\times\frac{\pi}{180}
  • 1^\circ = 60' = 3600''
  • s = r\theta
  • A = \tfrac12r^2\theta

Worked example

With the default numbers in the calculator above:

  1. In decimal degrees: 90 degrees
  2. In degrees, minutes and seconds: 90 deg 0 min 0 sec
  3. In radians: 1.5708 rad
  4. As a multiple of pi: pi/2 (0.5 pi)
  5. In gradians: 100 grad
  6. In turns: 0.25 turns
  7. In mils (NATO): 1600 mils
  8. The same direction between 0 and 360 degrees: 90 degrees
  9. Sine, cosine and tangent: sin = 1, cos = 0, tan = undefined
  10. Arc length on a circle of that radius: 1.5708
  11. Area of the sector: 0.785398

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: The angle you have is in Degrees, minutes and seconds (fill in the three boxes below).

  1. In decimal degrees: 40.4461 degrees
  2. In degrees, minutes and seconds: 40 deg 26 min 46 sec
  3. In radians: 0.705918 rad
  4. As a multiple of pi: 0.224701 pi
  5. In gradians: 44.9401 grad
  6. In turns: 0.11235 turns
  7. In mils (NATO): 719.042 mils
  8. The same direction between 0 and 360 degrees: 40.4461 degrees
  9. Sine, cosine and tangent: sin = 0.648733, cos = 0.761016, tan = 0.852455
  10. Arc length on a circle of that radius: 0.705918
  11. Area of the sector: 0.352959

Common questions

How do I convert degrees to radians?

Multiply by pi over 180. So 180 degrees is pi radians and 90 degrees is pi over 2.

How do I convert degrees minutes seconds to decimal degrees?

Add the minutes divided by 60 and the seconds divided by 3600 to the degrees.

What is a radian?

The angle at the centre of a circle that cuts off an arc equal in length to the radius.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any school mathematics text (angle measure).

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