# Degrees and radians

Radian is a measure of an angle, indicating the relationship between the arc length and the radius of a circle:

Since the arc length of a full circle is the same as the circumference of a circle given by:

a full circle in radians is given by:

A full circle is 360^{0} in degrees.

## Converting from degrees to radians

Decrees are converted to radians in the following way:

## Converting from radians to degrees

Radians are converted to degrees in the following way:

## Convert between degrees and radians

## Angles of more than 360 degrees and less than 0.

Sometimes you'll find angles in radians that are more than 360^{0} and less than 0^{0}.

That is, the angle in a circle is making more than a full rotation, just like when you turn around yourself several times.

However, it’s only the remaining angle after the last full rotation in the circle that is of significance.

In the following example we will show you how to find it.

A given angle is 810^{0}.

We need to find out, how many full rotations in a circle you can make. Since a full rotation is 360^{0} you have to divide 810 by 360. That makes 2 and the remainder 90.

So you can make 2 full rotations in a circle and you get the remainder angle 90^{0}.

Therefore 810^{0} is considered to be the same angle as 90^{0}.

## Locations on Earth – degrees, minutes and seconds

Distances on Earth can be measured in degrees instead of ordinary length units like miles and km. If you, for instance, make a journey the whole way around the Earth, you will travel 360^{0}.

In navigation on the sea a specific location on Earth is represented with latitude and longitude, which are usually denoted with degrees, minutes and second. One degree is divided into 60 minutes, which is divided into 60 seconds.

The location of the Empire State building in New York City is given by: 40^{0}44’54.36”N and 73^{0}59’8.5”W.

The first number is the latitude, which is 40 degrees, 44 minutes and 54.36 seconds. That is the number of degrees, minutes and seconds the Empire State building is from the Earth’s equator, where the latitude is 0. The N means that it’s northern latitudes, i.e. we are in the hemisphere to the north of the equator.

The second number is the longitude, which is 73 degrees, 59 minutes and 8.5 seconds. That is the number of degrees, minutes and seconds the Empire State building is from the World’s Prime Meridian, which is the north-south line passing through Greenwich in England, where the longitude is 0. The W means that it’s western latitudes, i.e. we are in the hemisphere to the west of the Greenwich line.

## Decimal degrees

Sometimes it’s more practical to use decimal degrees, which are a numbers with decimals, instead of using degrees, minutes and seconds.

The conversion between the two ways of representing degrees is given by:

So the latitude of The Empire State Building in decimal degrees is:

And the longitude is: