Angle calculator
A radian is the angle subtended by an arc equal in length to the radius, which makes 2π radians a full turn. Degrees are an arbitrary 360 inherited from Babylonian astronomy; radians are not arbitrary at all. That is why calculus is written in radians — the derivative of sin x is cos x only if x is in radians, and picks up an awkward factor of π/180 otherwise.
180 degrees is π radians, 200 gradians and half a turn. To convert degrees to radians multiply by π/180; to go back multiply by 180/π. A 45 degree angle is 0.7854 radians.
How to convert angles
Gradians are the unit almost nobody uses and they exist for a reason: a right angle is exactly 100 gradians and a full turn 400, which makes surveying arithmetic decimal throughout. They survive in some European surveying equipment and in a key on most scientific calculators that is pressed only by accident. Arcminutes and arcseconds. Sixtieths and three-thousand-six-hundredths of a degree. Remain genuinely standard in astronomy and navigation.
Questions
Multiply by π and divide by 180. Ninety degrees is π/2, about 1.5708.
2π, roughly 6.2832.
A hundredth of a right angle, so 400 to a full turn. It makes surveying arithmetic decimal.
A sixtieth of a degree. The human eye resolves about one arcminute, which is where display density limits come from.
Almost always the wrong angle mode. Trigonometric functions need to know whether you mean degrees or radians.