Click the Measure button to display the measure tool bar.  

To measure a distance 
click the "Distasnce" button and then click on 2 or more vertexes along the path you want to measure.  Click the Done button when you are finished and the distance will be labeled on the map.  Before clicking done you can change the units to feet, meters, or a few other choices.

To measure an area 
click the "Area" button and then click on several vertices around the permitter of the area.  Click the Done button when finished.  If you want to change the units of measurement do so before clicking done.

To clear the measurements 
from the map click the button.

Automatic Calculations
You can also add a special field to your layers so that anytime you click on a feature it will calculate and display the length or area of each feature.  See Add/Remove Fields for more information.

Video on this Topic
See Video:  Measureing Distances on Diamond Maps