Question 1

How do I convert degrees to radians?

Accepted Answer

Multiply by π/180. A right angle, 90°, becomes π/2 ≈ 1.5708 rad; a full turn, 360°, becomes 2π ≈ 6.2832 rad. As a single factor, 1° = 0.0174533 rad.

Question 2

How do I convert radians to degrees?

Accepted Answer

Multiply by 180/π, which is 57.2957795 — so one radian is a little over 57°. That awkward number is the price of a unit defined by the circle itself rather than by counting.

Question 3

Why do radians exist at all, if degrees are easier to read?

Accepted Answer

Because a radian is defined by the circle rather than imposed on it: one radian is the angle that cuts off an arc equal to the radius. That makes arc length simply r × θ, and it makes the calculus behave — the derivative of sin x is cos x only when x is in radians. In degrees you would carry a π/180 through every step.

Question 4

Where did 360 degrees come from?

Accepted Answer

From Babylonian base-60 arithmetic, and it stuck because 360 divides cleanly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more. That divisibility is genuinely useful: a third of a circle is a whole 120°, where a third of 400 gradians is not.

Question 5

What is a gradian, and who uses it?

Accepted Answer

A gradian, also written gon, is one hundredth of a right angle — 400 to a full turn, so 1 gon = 0.9°. It was a metric-era attempt to decimalise the circle. It survives mainly in surveying and civil engineering in parts of Europe, where a quarter turn being exactly 100 units simplifies bearing arithmetic.

Question 6

What are arcminutes and arcseconds?

Accepted Answer

Subdivisions of a degree in base 60: one degree is 60 arcminutes, one arcminute is 60 arcseconds, so a degree holds 3,600 arcseconds. They are written 12° 34′ 56″. Astronomy and navigation still use them because the units map onto real distance on the ground.

Question 7

How much ground does an arcsecond cover?

Accepted Answer

One arcminute of latitude is one nautical mile, 1,852 metres — that is where the nautical mile comes from. So one arcsecond of latitude is about 31 metres, and a tenth of an arcsecond is around 3 metres. It is why GPS coordinates quoted in DMS need decimal places on the seconds to be useful for anything smaller than a building.

Question 8

How do I read a coordinate like 51° 30′ 26″ N?

Accepted Answer

Degrees, then arcminutes, then arcseconds. Convert to decimal degrees by dividing: 30 ÷ 60 = 0.5, 26 ÷ 3600 = 0.00722, giving 51.50722° N. Mapping software almost always wants the decimal form, while charts and older references use DMS.

Question 9

What is a turn or revolution?

Accepted Answer

One full rotation: 360°, 2π radians, or 400 gradians. Mechanical and rotational work often quotes angles in turns or in revolutions per minute rather than in any of the three, because the whole rotation is the natural unit there.

Question 10

Why does my converted value have so many decimals?

Accepted Answer

Because π is irrational, so almost every conversion between degrees and radians is an approximation somewhere. Keep the extra digits through intermediate steps and round only at the end — rounding 57.2957795 to 57.3 early will show up as visible error over a long distance or a long calculation.

Angle of latitude	Distance on Earth
1 degree	≈ 111 km
1 arcminute (1′)	1 nautical mile — 1,852 m
1 arcsecond (1″)	≈ 31 m
0.1 arcsecond	≈ 3 m

From	To	Value
180°	Radians	π rad (3.14159 rad)
90°	Radians	π/2 rad (1.5708 rad)
1 Radian	Degrees	57.2958°
1 Gradian	Degrees	0.9°
1 Arcminute	Degrees	0.0166667°
1 Arcsecond	Degrees	0.00027778°

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