Degrees to Radians: The One Multiplication Behind Any Angle

A protractor reads degrees, a scientific calculator's trig functions expect radians, some European surveying instruments use gradians, and a wrench spec or a spinning wheel is measured in full turns. These are all the same kind of quantity — an angle — just scaled by different fixed multiples, so converting between them is one multiplication, done through a common base unit.

The method: convert through a fixed base unit

This tool uses degrees as the base. Every other angle unit is a fixed number of degrees — a radian is always about 57.29577951 degrees, a full turn is always 360 degrees, by definition. To convert a value from one unit to another: multiply by the source unit's degrees-per-unit, then divide by the target unit's degrees-per-unit.

answer = value × (source degrees-per-unit) ÷ (target degrees-per-unit)

That's the whole method. Degrees to radians, gradians to degrees, arcminutes to degrees — all the same formula, just a different pair of numbers.

The numbers

• 1 degree = 1 degree (the base)
• 1 radian = 57.29577951 degrees (180 ÷ π, since a half turn is π radians and 180°)
• 1 gradian = 0.9 degrees (a right angle is 100 gradians, and also 90 degrees)
• 1 arcminute = 1/60 of a degree (0.016666… degrees)
• 1 turn = 360 degrees (one full rotation)

A worked example: 45 degrees in radians

Divide by a radian's 57.29577951 degrees, since degrees is already the base: 45 ÷ 57.29577951 = 0.785398 radians. That's a known value worth checking against — 45 degrees is one-eighth of a full turn, and π÷4 (the textbook way to write it) is 0.7853981633…, matching to six decimal places.

A second example: a full turn, back to radians

360 ÷ 57.29577951 = 6.283185 radians — consistent with the textbook fact that one full turn is 2π radians (6.2831853…), off only in the eighth decimal place because the stored 57.29577951 is π's reciprocal truncated to eight decimals rather than carried out further — the same kind of harmless truncation noise other conversion tables in this series have flagged, not an error in the method.

A third example: gradians, the metric angle

A right angle is 100 gradians by design, so 30 gradians should be more than a quarter of that: 30 × 0.9 degrees per gradian = 27 degrees, an exact result with no rounding, since 0.9 is an exact decimal. Going the other way, a 180-degree straight line is 180 ÷ 0.9 = 200 gradians — also exact, and the reason gradians exist at all: a right angle, a straight line and a full turn all come out as clean round numbers (100, 200, 400) in that system.

A fourth example: arcminutes, for small angles

Navigation and telescope pointing often specify angles in degrees and arcminutes, the way a clock face splits an hour into minutes. An offset of 15 arcminutes: 15 × (1/60) degree per arcminute = 0.25 degrees — small enough that a compass bearing or a star's position needs the finer unit to be useful at all.

A fifth example: a wheel spinning 2.5 turns

2.5 × 360 degrees per turn = 900 degrees. A wheel that has turned 900 degrees has gone around two and a half times; subtracting 720 (two full turns) leaves 180 degrees, meaning it now faces exactly the opposite direction from where it started — the everyday version of the same multiply-and-divide method, just tracking a rotation instead of a static angle.

What the tool adds

The numbers above are all you need, on paper, for any pair of angle units. The tool's job is just speed and fewer mistyped digits: pick the two units, type the value, and it runs the same multiplication instantly across any of the five units it knows, entirely on your device.