Degrees
Definition: A measure of an
angle.
One degree is one 360th part of a full circle.
Try this
Adjust the angle below by dragging the orange at R. Note the number of degrees for any particular angle.
Measure of an angle
In geometry, an
angle.
is measured in degrees, where a full circle is 360 degrees. A small angle might be around 30 degrees.
Usually, when a finer measure is needed we just add decimal places to the degrees. For example 45.12°
The small circle after the number means "degrees". So the above would be pronounced "forty five point one two degrees".
Degrees  Minutes  Seconds
When measuring things like latitude and longitude, each degree is divided into minutes and seconds.
The degree is divided in to 60 minutes. For even finer measurements the minute is divided again into 60 seconds,
However this last measure is so small, it only used where angles are
subtended
over extreme distances such as astronomical measurements, and measuring latitude and longitude.
These minutes and seconds have (confusingly) nothing to do with time. They are just smaller and smaller parts of a degree.
See also Degrees  Minutes  Seconds calculator
for a calculator that can add and subract angles in this form.
Unit 
Written 
Pronounced 
Degrees 
With a small circle after the number. Example 61° 
"61 degrees" 
Minutes 
With a small dash after the number. Example 34° 21' 
"34 degrees, 21 minutes" 
Seconds 
With two small dashes. Example 32° 34' 44'' 
"32 degrees, 34 minutes, 44 seconds" 
When minutes and seconds are used alone, we usually say "arc minutes" and "arc seconds" to avoid confusion with time units.
Which direction to measure?
In the figure above, adjust the point R so the line crosses the point marked 315°. Starting at Q and going counterclockwise we see the measure is 315°. But if we were to go clockwise from Q it would be 45° (360315). Which is correct?
They both are, but by convention the smaller one is assumed. That is why the angle at the center shows 45° under these circumstances.
The larger measure (315°) is called the
reflex angle RPQ.
Angles you should know
Use the figure above to become familiar with what various angle measures look like when measured in degrees. In general, you should be able
to visually estimate any angle to within about 15°, and you should be able to recognize the common angles (shown in red) on sight and sketch them yourself.
Other measures

Radians
An angle can be measured in radians where the full circle is 2 pi radians (about 6.28).
This is used extensively in trigonometry.

Grads
In some surveying work the grad is used. There are 400 grads in a circle and so a right angle is 100 grads. You will rarely see this unit. Think of grads as 'metric degrees'.

Nautical angles
Ship's navigators use angles that are measured slightly differently, using a system designed hundreds of years ago for the Nautical Alamanac  a book of navigation tables.
Each degree is divided into 60 minutes as usual, but there are no seconds.
The minute is expressed as a decimal instead. For example 23° 34.62' is read as "23 degrees 34.62 minutes.
See also Nautical Angle Calculator".
Things to try
 In the figure above, click on 'hide details'.
 Adjust the position of the point R
 Estimate the measure of the angle RPQ
 Click 'show details' to see how close you got
 Repeat.
You should especially be able to estimate angles close to the red ones in the figure above, since they appear frequently in geometry.
Other angle topics
General
Angle Types
Angle relationships
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